International A LevelMaths: Statistics 2EdexcelExam Questions1. Statistical Distributions1.3 Continuous Random Variables

Continuous Random Variables (Edexcel International A Level Maths: Statistics 2)

Exam Questions

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1
Sme Calculator4 marks

If straight f left parenthesis x right parenthesis  is the probability density function for a random variable, X then the area under the graph of the probability density function must equal 1.

State, with a reason, whether each of the following could be graphs of probability density functions.

q1a-1-3-ial-s2-crv-easy-satistics-2

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2
Sme Calculator6 marks

If straight f left parenthesis x right parenthesis is the probability density function for a random variable, X which takes values in the interval open parentheses a comma space b close parentheses then:

integral subscript a superscript b straight f open parentheses x close parentheses d x equals 1

Use integration to decide whether the following functions could represent probability distribution functions.

(i)
straight f open parentheses x close parentheses equals open curly brackets table row cell 2 x end cell cell 0 less or equal than x less or equal than 1 end cell row 0 cell otherwise. end cell end table close

(ii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 1 third x squared end cell cell 1 less or equal than x less or equal than 4 end cell row 0 cell otherwise. end cell end table close

(iii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 4 over 27 open parentheses x cubed plus 1 close parentheses end cell cell negative 1 less or equal than x less or equal than 2 end cell row 0 cell otherwise. end cell end table close

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3a
Sme Calculator2 marks

The continuous random variable, X, has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell 1 over 24 open parentheses x cubed plus 2 close parentheses end cell cell 1 less or equal than x less or equal than 3 end cell row 0 cell otherwise. end cell end table close

The value ofspace space P left parenthesis a less than X less than b right parenthesis is equal to

integral subscript a superscript b straight f open parentheses x close parentheses d x.

Findspace space P left parenthesis 1 less than X less than 2 right parenthesis.

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3b
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(i)
Explain why P left parenthesis X greater than 2.5 right parenthesis equals integral subscript 2.5 end subscript superscript 3 1 over 24 open parentheses x cubed plus 2 close parentheses d x.
(ii)
Hence find P left parenthesis X greater than 2.5 right parenthesis.

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3c
Sme Calculator2 marks

State, with a reason, the value of

(i)
P left parenthesis 5 less than X less than 10 right parenthesis
(ii)
P left parenthesis X equals 2 right parenthesis.

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4
Sme Calculator4 marks

The graph below shows the probability density function of a continuous random variable, X.

q4a-1-3-ial-s2-crv-easy-satistics-2

By finding relevant areas under the graph, find:

(i)
P left parenthesis 0 less than X less than 0.1 right parenthesis
(ii)
P left parenthesis X greater than 0.15 right parenthesis
(iii)
P left parenthesis 0.05 less than X less than 0.12 right parenthesis.

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5a
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Evaluate the following definite integral, giving your answer in terms of k.

integral subscript 3 superscript 9 k open parentheses x squared plus 3 close parentheses d x

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5b
Sme Calculator2 marks

Hence find the value of k given that straight f open parentheses x close parentheses is a probability density function where

straight f open parentheses x close parentheses equals open curly brackets table row cell k open parentheses x squared plus 3 close parentheses end cell cell 3 less or equal than x less or equal than 9 end cell row 0 cell otherwise. end cell end table close

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6a
Sme Calculator2 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell 3 over 16 open parentheses x squared minus 1 close parentheses end cell cell 2 less or equal than x less or equal than 3 end cell row 0 cell otherwise. end cell end table close

Find  integral 3 over 16 x open parentheses x squared minus 1 close parentheses d x.

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6b
Sme Calculator1 mark

Hence find the value of straight E left parenthesis X right parenthesis using the formula

straight E left parenthesis X right parenthesis equals integral x straight f open parentheses x close parentheses straight d x.

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6c
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Find integral 3 over 16 x squared open parentheses x squared minus 1 close parentheses straight d x.

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6d
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Hence find the value of straight E left parenthesis X ² right parenthesis using the formula

straight E left parenthesis X ² right parenthesis equals integral x squared straight f open parentheses x close parentheses straight d x.

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6e
Sme Calculator1 mark

Hence find the value of Var left parenthesis X right parenthesis space spaceusing the formula Var left parenthesis X right parenthesis equals straight E left parenthesis X squared right parenthesis minus left parenthesis straight E left parenthesis X right parenthesis right parenthesis ².

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7
Sme Calculator4 marks

The mode of a continuous random variable X, where it exists, is a value for X where the probability density function is at its maximum.

State the mode, where it exists, of the continuous random variables which have probability density functions show in the graphs below.

q9a-1-3-ial-s2-crv-easy-satistics-2-2

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8
Sme Calculator6 marks

The median, m, of a continuous random variable X, is the value for X which splits the under the graph of the probability density function exactly in half, such that straight P left parenthesis X less than m right parenthesis equals 0.5.

Find the value of the median of the continuous random variables which have probability density functions show in the graphs below.

q10a-1-3-ial-s2-crv-easy-satistics-2

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9a
Sme Calculator1 mark

The continuous uniform (rectangular) distribution over the interval open parentheses a comma space b close parentheses has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell fraction numerator 1 over denominator b minus a end fraction end cell cell a less than x less than b end cell row 0 cell otherwise. end cell end table close

The random variable X follows a rectangular distribution over the interval left parenthesis 2 comma 18 right parenthesis.

The graph below shows the probability density function of X. Write down the values of a comma space b and c.

q9a1-1-3-ial-s2-crv-easy-satistics-2-2

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9b
Sme Calculator3 marks

Write down

(i)
the median of X
(ii)
the value of straight E left parenthesis X right parenthesis
(iii)
straight P left parenthesis 14 less than X less than 18 right parenthesis.

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10a
Sme Calculator3 marks

The continuous random variable X has probability density function straight f open parentheses x close parentheses given by

straight f open parentheses x close parentheses equals open curly brackets table row cell 1 fourth x end cell cell 0 less or equal than x less or equal than 2 end cell row cell 5 over 9 minus 1 over 9 x end cell cell 2 less or equal than x less or equal than 5 end cell row 0 cell otherwise. end cell end table close

Find P left parenthesis 1 less or equal than X less or equal than 4 right parenthesis by calculating

integral subscript 1 superscript 2 open parentheses 1 fourth x close parentheses d x plus integral subscript 2 superscript 4 open parentheses 5 over 9 minus 1 over 9 x close parentheses d x

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10b
Sme Calculator3 marks

Find straight E left parenthesis X right parenthesis by calculating

integral subscript 0 superscript 2 open parentheses 1 fourth x close parentheses left parenthesis x right parenthesis straight d x plus integral subscript 2 superscript 5 open parentheses 5 over 9 minus 1 over 9 x close parentheses left parenthesis x right parenthesis straight d x

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1
Sme Calculator6 marks

State, with a reason, whether each of the following could be graphs of probability density functions.

q1a-1-3-ial-s2-crv-medium-satistics-2

This is a semicircle.

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2
Sme Calculator6 marks

State, with a reason, whether the following functions could represent probability distribution functions.

(i)
straight f open parentheses x close parentheses equals open curly brackets table row cell 3 open parentheses x to the power of 5 minus 2 x cubed close parentheses end cell cell 1 less or equal than x less or equal than 2 end cell row 0 cell otherwise. end cell end table close

(ii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 3 over 8 open parentheses x plus 1 close parentheses squared end cell cell negative 1 less or equal than x less or equal than 1 end cell row 0 cell otherwise. end cell end table close

(iii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 6 over x squared end cell cell negative 3 less or equal than x less or equal than negative 2 end cell row 0 cell otherwise. end cell end table close

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3a
Sme Calculator2 marks

The continuous random variable, X, has probability density function

f open parentheses x close parentheses equals open curly brackets table row cell k open parentheses 3 x to the power of 4 minus 2 x close parentheses end cell cell 1 less or equal than x less or equal than 2 end cell row 0 cell otherwise. end cell end table close

Show that k equals 5 over 78.

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3b
Sme Calculator4 marks
(i)
Find P left parenthesis 1.2 less than X less than 1.8 right parenthesis.
(ii)
Find  P left parenthesis X less than 1.5 right parenthesis.

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4a
Sme Calculator2 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell k open parentheses 2 plus square root of x close parentheses end cell cell 4 less or equal than x less or equal than 9 end cell row 0 cell otherwise. end cell end table close

Show that k equals 3 over 68.

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4b
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Find straight E left parenthesis X right parenthesis.

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4c
Sme Calculator3 marks

Find Var left parenthesis X right parenthesis.

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5
Sme Calculator4 marks

Find the mode of the continuous random variable, X, which has probability density function defined by:

(i)
straight f open parentheses x close parentheses equals open curly brackets table row cell 2 over 21 x end cell cell 2 less or equal than x less or equal than 5 end cell row 0 cell otherwise. end cell end table close

(ii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 1 over 9 open parentheses 4 minus open parentheses x plus 1 close parentheses squared close parentheses end cell cell negative 2 less or equal than x less or equal than 1 end cell row 0 cell otherwise. end cell end table close

(iii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 1 half sin open parentheses x close parentheses end cell cell 0 less or equal than x less or equal than pi end cell row 0 cell otherwise. end cell end table close

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6a
Sme Calculator2 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell 5 over 62 square root of x cubed end root end cell cell 1 less or equal than x less or equal than 4 end cell row 0 cell otherwise. end cell end table close

Find an expression, in terms of m, for

integral subscript 1 superscript m 5 over 62 square root of x cubed end root d x.

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6b
Sme Calculator3 marks

Hence find the median of X.

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7a
Sme Calculator3 marks

The continuous random variable X follows a continuous uniform distribution over the interval left parenthesis 3 comma 28 right parenthesis so that its probability density function is

f open parentheses x close parentheses equals open curly brackets table row cell 1 over 25 end cell cell 3 less than x less than 28 end cell row 0 cell otherwise. end cell end table close

Find

(i)
P left parenthesis X less than 15 right parenthesis
(ii)
P left parenthesis 20 less than X less than 30 right parenthesis
(iii)
P left parenthesis X equals 25 right parenthesis.

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7b
Sme Calculator3 marks
(i)
Write down straight E left parenthesis X right parenthesis .
(ii)
Find Var left parenthesis straight X right parenthesis .

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8a
Sme Calculator2 marks

May calls her nephew Peter each day to check on him. May enjoys talking so much so that Peter has programmed his phone to disconnect from a call after 15 minutes. The length of a call, in minutes, between May and Peter is denoted by the continuous random variable T. It is modelled by the probability density function

straight f open parentheses t close parentheses equals open curly brackets table row cell 1 over 1125 t squared end cell cell 0 less or equal than t less or equal than 15 end cell row 0 cell otherwise. end cell end table close

Find the mean length of a call.

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8b
Sme Calculator3 marks

Find the standard deviation of the lengths of calls.

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8c
Sme Calculator2 marks

Find the probability that a random call between May and Peter will last longer than 10 minutes.

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8d
Sme Calculator1 mark

Out of the next 100 calls between May and Peter, estimate the number of them that will last less than 10 minutes.

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9a
Sme Calculator2 marks

The mass, in kilograms, of a wrestler in a local club is denoted by the continuous random variable M. It is modelled by the probability density function

straight f open parentheses m close parentheses equals open curly brackets table row cell k minus 1 over 9000 open parentheses m minus 80 close parentheses squared end cell cell 70 less than m less than 100 end cell row 0 cell otherwise. end cell end table close

Show that k equals 4 over 90.

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9b
Sme Calculator2 marks

Wrestlers who weigh less than 85 kg can enter Heavy Middleweight competitions.

Find the probability that a randomly selected wrestler can enter Heavy Middleweight competition.

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9c
Sme Calculator3 marks

Wrestlers who weigh less than 80 kg can enter Middleweight competitions.

Given that a wrestler can enter Heavy Middleweight competitions, find the probability that they can also enter Middleweight competitions.

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10a
Sme Calculator4 marks

The continuous random variable X has probability density function straight f open parentheses x close parentheses given by

straight f open parentheses x close parentheses equals open curly brackets table row cell 1 fifth open parentheses x minus 1 close parentheses end cell cell 1 less or equal than x less or equal than k end cell row cell 2 minus 2 over 5 x end cell cell k less than x less or equal than 4 end cell row 0 cell otherwise. end cell end table close

Find the value of k.

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10b
Sme Calculator2 marks

Find P left parenthesis X less than 3.5 right parenthesis.

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10c
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Show that straight E left parenthesis X right parenthesis equals 3.

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1
Sme Calculator5 marks

State, with a reason, whether the following functions could represent probability distribution functions.

(i)
straight f open parentheses x close parentheses equals open curly brackets table row cell 1 third open parentheses x squared minus 2 close parentheses end cell cell 1 less or equal than x less or equal than 2 end cell row 0 cell otherwise. end cell end table close

(ii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 4 over x to the power of 5 end cell cell x greater or equal than 1 end cell row 0 cell otherwise. end cell end table close

(iii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 2 over x squared end cell cell negative 1 less or equal than x less or equal than 2 end cell row 0 cell otherwise. end cell end table close

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2a
Sme Calculator4 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table attributes columnalign left columnspacing 1.4ex end attributes row cell begin inline style 1 fourth end style x open parentheses x minus 1 close parentheses open parentheses x plus 1 close parentheses end cell cell 1 less or equal than x less or equal than k end cell row 0 cell otherwise. end cell end table close

Find the exact value for k .

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2b
Sme Calculator4 marks

Find P left parenthesis X less than E left parenthesis X right parenthesis space right parenthesis.

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3a
Sme Calculator4 marks

Harry is trying to draw specific lengths without measuring equipment. He draws a straight line and stops when he thinks it is 10 cm long. The actual length of Harry’s line, L cm, can be modelled by the probability density function

straight f open parentheses l close parentheses equals open curly brackets table row cell 20 over 421 open parentheses l minus 10.5 close parentheses to the power of 4 end cell cell 8 less or equal than l less or equal than 12 end cell row 0 cell otherwise. end cell end table close

Estimate the lower quartile of the lengths of Harry’s lines.

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3b
Sme Calculator1 mark

Harry tries to draw a 10 cm line 80 times.

Write down how many of Harry’s lines you would expect to be less than the lower quartile.

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4a
Sme Calculator1 mark

The diagram below shows the probability density function, straight f left parenthesis x right parenthesis, of a random variable X. straight f left parenthesis x right parenthesis equals k when 0 less or equal than x less or equal than a, otherwise straight f left parenthesis x right parenthesis equals 0.

q4a-1-3-ial-s2-crv-hard-satistics-2

Write down an expression for k  in terms of a.

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4b
Sme Calculator1 mark

Write down an expression for straight E left parenthesis X right parenthesis in terms of a.

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4c
Sme Calculator3 marks

Find a simplified expression for Var left parenthesis X right parenthesis in terms of a.

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4d
Sme Calculator2 marks

Given thatspace space P left parenthesis X greater than 5 right parenthesis equals 0.6 spacefind the value of a.

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5
Sme Calculator6 marks

Find the mode of the continuous random variable, X, which has probability density function defined by:

(i)
straight f open parentheses x close parentheses equals open curly brackets table row cell 3 over 140 open parentheses 5 minus 6 x minus x squared close parentheses end cell cell negative 4 less or equal than x less or equal than 0 end cell row 0 cell otherwise. end cell end table close

(ii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 3 over 34 open parentheses 10 minus open parentheses x minus 2 close parentheses squared close parentheses end cell cell 3 less or equal than x less or equal than 5 end cell row 0 cell otherwise. end cell end table close

(iii)
straight f open parentheses x close parentheses equals open curly brackets table row cell 1 half open parentheses sin open parentheses x close parentheses plus cos open parentheses x close parentheses close parentheses end cell cell 0 less or equal than x less or equal than pi over 2 end cell row 0 cell otherwise. end cell end table close

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6a
Sme Calculator1 mark

The diagram below shows the probability density function, straight f left parenthesis x right parenthesis, of a random variable X.

q6a-1-3-ial-s2-crv-hard-satistics-2

Find the value of k.

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6b
Sme Calculator2 marks

Find straight P left parenthesis X greater than 0.4 right parenthesis.

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6c
Sme Calculator4 marks

Find the median of X.

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7a
Sme Calculator2 marks

The diagram below shows the probability density function, straight f open parentheses t close parentheses, of a random variable T.

q7a-1-3-ial-s2-crv-hard-satistics-2

For a less or equal than t less or equal than b comma space space straight f left parenthesis t right parenthesis equals 3 over 32 left parenthesis 10 t minus t squared minus 21 right parenthesis comma elsewhere straight f left parenthesis t right parenthesis equals 0.

Find the values of a and b.

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7b
Sme Calculator4 marks

State the value of straight E left parenthesis T right parenthesis and find Var left parenthesis T right parenthesis.

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7c
Sme Calculator1 mark

Find straight P left parenthesis negative 2 less or equal than T less or equal than 5 right parenthesis.

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7d
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Given that P left parenthesis 4 less or equal than T less or equal than 6 right parenthesis equals 11 over 16 find P left parenthesis T greater or equal than 6 right parenthesis.

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8a
Sme Calculator2 marks

Geoff is taking part in a quiz where he has 4 seconds to answer each question. The time, in seconds, it takes Geoff to answer a question is denoted T which can be modelled by the probability density function, straight f left parenthesis t right parenthesis comma  shown below.

q8a-1-3-ial-s2-crv-hard-satistics-2

Find the probability that Geoff answers a question in less than 2 seconds.

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8b
Sme Calculator4 marks

Find the mean time it takes Geoff to answer a question.

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8c
Sme Calculator3 marks

There are 10 questions in the quiz. Find the probability that for at least one of the questions Geoff takes longer than 3 seconds.

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1a
Sme Calculator2 marks

The diagram below shows the probability density function, straight f left parenthesis x right parenthesis, of a random variable X.

q1a-1-3-ial-s2-crv-very-hard-satistics-2

Find the value of k.

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1b
Sme Calculator2 marks

Find P left parenthesis X less or equal than 6 right parenthesis.

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1c
Sme Calculator4 marks

Find the interquartile range of X.

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2a
Sme Calculator2 marks

A computer takes T seconds to start up. The random variable, T, can be modelled by the probability distribution, straight f left parenthesis t right parenthesis, shown below.

q2a-1-3-ial-s2-crv-very-hard-satistics-2

For a less or equal than t less or equal than b comma space space straight f left parenthesis t right parenthesis equals 1 over 8 pi square root of left parenthesis 12 t minus t squared minus 20 right parenthesis end root comma elsewhere straight f left parenthesis t right parenthesis equals 0.

Find the values of a and b.

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2b
Sme Calculator1 mark

State the mean time for the computer to start up.

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2c
Sme Calculator1 mark

Find probability that the computer takes between 6 and 15 seconds to start up.

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2d
Sme Calculator2 marks

Given that P left parenthesis 4 less or equal than T less or equal than 8 right parenthesis equals fraction numerator square root of 3 over denominator 2 pi end fraction plus 1 third comma find the exact value of P left parenthesis T greater or equal than 8 right parenthesis.

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3a
Sme Calculator1 mark

The diagram below shows the probability density function, g left parenthesis y right parenthesis, of a random variable Y.

q3a-1-3-ial-s2-crv-very-hard-satistics-2

Find the value of k.

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3b
Sme Calculator3 marks

Find P left parenthesis 5 less than Y less than 6 right parenthesis.

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3c
Sme Calculator2 marks

Find the median of Y.

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3d
Sme Calculator4 marks

Find straight E left parenthesis Y right parenthesis.

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4a
Sme Calculator5 marks

The random variable, U, follows a continuous uniform distribution on the interval open square brackets a comma b close square brackets. The probability density function  is defined by:

g open parentheses u close parentheses equals open curly brackets table row k row 0 end table close space space space space space space space space space table row cell a less or equal than u less or equal than b end cell row cell otherwise. end cell end table

(i)
Write down an expression for k in terms of a and b.

(ii)
Write down an expression for straight E left parenthesis U right parenthesis in terms of a and b.

(iii)
Show that  Var open parentheses U close parentheses equals open parentheses b minus a close parentheses squared over 12..

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4b
Sme Calculator3 marks

Given that straight E left parenthesis U right parenthesis equals 8.5 spaceand Var left parenthesis U right parenthesis equals 6.75, find the values of a and b.

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5a
Sme Calculator6 marks

A meeting at a company lasts T hours. The random variable T can be modelled by the probability density function

straight f open parentheses t close parentheses equals open curly brackets table row cell 3 over 4 t open parentheses t minus 2 close parentheses squared end cell row 0 end table close space space space space space space space space space space space space table row cell 0 less or equal than t less or equal than 2 end cell row cell otherwise. end cell end table

Find the mean time of a meeting and find the standard deviation of times.

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5b
Sme Calculator4 marks

Show that the median length of a meeting is 0.771 hours, correct to 3 significant figures.

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5c
Sme Calculator4 marks

By using differentiation, find the mode of the times.

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6a
Sme Calculator3 marks

The continuous random variable X has probability density function

straight f open parentheses x close parentheses equals open curly brackets table row cell k over x cubed end cell row 0 end table close space space space space space space space space space space space space space space space space space table row cell 2 less or equal than x less or equal than a end cell row cell otherwise. end cell end table

Show that k equals fraction numerator 8 a squared over denominator a squared minus 4 end fraction.

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6b
Sme Calculator5 marks

Given that straight E left parenthesis X right parenthesis equals 20 over 7 comma show that a equals 5 and hence find the value of k.

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6c
Sme Calculator3 marks

Find the exact value of Var left parenthesis X right parenthesis .

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7a
Sme Calculator1 mark

Paul is travelling around France to try to find the perfect baguette. The random variable C represents the cost, in euros, of a baguette. It can be modelled by the probability density function

straight f open parentheses c close parentheses equals open curly brackets table row cell 1 over 36 open parentheses 14 c minus c squared minus 40 close parentheses end cell row 0 end table close space space space space space space space space space space space space space space space space table row cell 4 less or equal than c less or equal than 10 end cell row cell otherwise. end cell end table

Sketch the graph of y equals straight f left parenthesis c right parenthesis.

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7b
Sme Calculator1 mark

Explain why the mean price of a baguette is €7.

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7c
Sme Calculator2 marks

Given that P left parenthesis C less than 9 right parenthesis equals 25 over 27 comma find the value of P left parenthesis 5 less than C less than 9 right parenthesis.

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8a
Sme Calculator4 marks

The continuous random variable X has probability density function straight f left parenthesis x right parenthesis given by

straight f open parentheses x close parentheses equals open curly brackets table row cell 1 over 7 x to the power of 4 end cell cell space space space minus 1 less or equal than x less or equal than 1 end cell row cell 1 over 20 x cubed minus 1 over 35 x end cell cell space space space space space space 1 less than x less than 3 end cell row k cell space space space space space 3 less or equal than x less or equal than 6 end cell row 0 cell space space space space space space space otherwise end cell end table close

where k is a constant.

Find the exact value of k.

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8b
Sme Calculator2 marks

Find P left parenthesis X greater than 0 right parenthesis.

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8c
Sme Calculator4 marks

Find straight E left parenthesis X right parenthesis.

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