Cumulative Distribution Function (Edexcel International A Level Maths: Statistics 2)

Exam Questions

4 hours24 questions
1a
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5 marks

The continuous random variable, X, has a cumulative distribution function straight F left parenthesis x right parenthesis given by

F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 1 end cell row cell 1 over 16 open parentheses x minus 1 close parentheses squared end cell cell 1 less or equal than x less or equal than 5 end cell row 1 cell x greater than 5 end cell end table close

Find:

(i)
straight F left parenthesis 0 right parenthesis
(ii)
straight F left parenthesis 1 right parenthesis
(iii)
straight F left parenthesis 4 right parenthesis
(iv)
straight F left parenthesis 5 right parenthesis
(v)
straight F left parenthesis 9 right parenthesis
1b
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4 marks

Write the following in terms of straight F left parenthesis a right parenthesis and straight F left parenthesis b right parenthesis.

(i)
P left parenthesis X less than a right parenthesis
(ii)
P left parenthesis X less or equal than a right parenthesis
(iii)
P left parenthesis X greater than a right parenthesis
(iv)
P left parenthesis a less than X less than b right parenthesis

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2a
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1 mark

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 attributes columnalign left columnspacing 1.4ex end attributes row cell 3 over 19 x squared 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

The cumulative distribution function straight F open parentheses x close parentheses is given by

straight F open parentheses x close parentheses equals open curly brackets table row a cell x less than 2 end cell row cell g open parentheses x close parentheses end cell cell 2 less or equal than x less or equal than 3 end cell row b cell x greater than 3 end cell end table close

where g left parenthesis x right parenthesis is a function a and b are constants.

Explain why space a equals 0 space and write down the value of b.

2b
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2 marks

Use integration to show that

g left parenthesis x right parenthesis equals k x to the power of n plus c

where k are n constants to be found and c is a constant of integration.

2c
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2 marks

Use g left parenthesis 2 right parenthesis equals 0 to find the value of c. Verify this by checking that space g left parenthesis 3 right parenthesis equals 1.

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3a
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1 mark

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 0 end cell row cell 1 over 10 x cubed plus 1 over 10 x end cell cell 0 less or equal than x less or equal than 2 end cell row 1 cell x greater than 2 end cell end table close

The probability density function is given by

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

where g left parenthesis x right parenthesis is a function and a is a constant.

Write down the value of a.

3b
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2 marks

Use differentiation to find g left parenthesis x right parenthesis.

3c
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2 marks

Sketch a graph of straight f left parenthesis x right parenthesis.

3d
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1 mark

Use the graph to write down the mode of X.

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4a
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2 marks

The continuous random variable Y is uniformly distributed over the interval open square brackets 3 comma 8 close square brackets.

Find:

(i)
straight E left parenthesis Y right parenthesis
(ii)
Var left parenthesis Y right parenthesis.
4b
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2 marks

Specify fully the probability density function straight f left parenthesis y right parenthesis.

4c
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2 marks

Specify fully the cumulative distribution function straight F left parenthesis y right parenthesis.

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5a
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2 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 x squared end cell cell 0 less than x less than 1 end cell row cell 1 minus 1 third x 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 cumulative distribution function straight F left parenthesis x right parenthesis is given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less or equal than 0 end cell row cell g open parentheses x close parentheses end cell cell 0 less than x less than 1 end cell row cell h open parentheses x close parentheses end cell cell 1 less or equal than x less or equal than 3 end cell row 1 cell x greater than 3 end cell end table close

where g left parenthesis x right parenthesis and h left parenthesis x right parenthesis are functions.

Explain why

(i)
g left parenthesis 0 right parenthesis equals 0
(ii)
h left parenthesis 3 right parenthesis equals 1
(iii)
g left parenthesis 1 right parenthesis equals h left parenthesis 1 right parenthesis.
5b
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4 marks

Use integration and part (a) to find g left parenthesis x right parenthesis and h left parenthesis x right parenthesis.

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6a
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2 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 2 end cell row cell 1 fourth x squared minus x plus 1 end cell cell 2 less or equal than x less or equal than 3 end cell row cell 3 over 8 x minus 7 over 8 end cell cell 3 less than x less or equal than 5 end cell row 1 cell x greater than 5 end cell end table close

The median of X is denoted by m.

(i)
Find straight F left parenthesis 3 right parenthesis  and straight F left parenthesis 5 right parenthesis.
(ii)
Explain why m satisfies the inequality 3 less than m less than 5.
6b
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2 marks
(i)
Explain why m satisfies the equation

3 over 8 m minus 7 over 8 equals 1 half.

(ii)
Hence find the median of X.
6c
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4 marks

Using differentiation, find the probability density function, straight f left parenthesis x right parenthesis, of the random variable X for all values of x.

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1a
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5 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 2 end cell row cell 1 over 8 open parentheses x minus 2 close parentheses cubed end cell cell 2 less or equal than x less or equal than 4 end cell row 1 cell x greater than 4 end cell end table close

Find

(i)
P left parenthesis X less or equal than 2.5 right parenthesis
(ii)
P left parenthesis 2.9 less or equal than X less or equal than 3.1 right parenthesis
(iii)
P left parenthesis X greater than 3 right parenthesis
(iv)
P left parenthesis X equals 4 right parenthesis.
1b
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2 marks
(i)
Find the value of k such that P left parenthesis X less than k right parenthesis equals 0.5.
(ii)
Hence write down the name of the average represented by k.

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2a
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3 marks

The continuous random variable, X, has a cumulative distribution function straight F left parenthesis x right parenthesis given by

F left parenthesis x right parenthesis equals open curly brackets table row 0 cell x less than 0 end cell row cell 3 over 4000 x cubed plus 1 over 400 x ² end cell cell 0 less or equal than x less or equal than 10 end cell row 1 cell x greater than 10 end cell end table close

Verify that the median of X lies between x equals 7.7 and 7.8.

2b
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3 marks

Specify fully the probability density function straight f left parenthesis x right parenthesis .

2c
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2 marks

By sketching straight f left parenthesis x right parenthesis, show that the mode of X is 10.

2d
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3 marks

Show that E left parenthesis X right parenthesis equals 175 over 24.

2e
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1 mark

Explain why the distribution of X is negatively skewed.

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3a
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3 marks

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

f open parentheses x close parentheses equals open curly brackets table attributes columnalign left end attributes row cell k x to the power of 4 end cell row 0 end table close space space space space space space space space table row cell negative 2 less or equal than x less or equal than 3 end cell row cell otherwise. end cell end table

Show that k equals 1 over 55.

3b
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3 marks

Find, in full, the cumulative distribution function straight F left parenthesis x right parenthesis.

3c
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2 marks

Find the lower quartile of X, give your answer to 3 significant figures.

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4a
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3 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

F left parenthesis x right parenthesis equals open curly brackets table row 0 cell x less than 5 end cell row cell fraction numerator x minus 5 over denominator 7 end fraction end cell cell 5 less or equal than x less or equal than 12 end cell row 1 cell x greater or equal than 12 end cell end table close

Find the probability density function straight f left parenthesis x right parenthesis .

4b
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1 mark

Write down the name of the distribution of X .

4c
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3 marks

Find the mean and the variance of X.

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5a
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2 marks

The continuous random variable T represents the times, in hours, that children at a boarding school take to tidy their rooms. The cumulative distribution function of T is

F left parenthesis t right parenthesis equals open curly brackets table row 0 cell t less than 0 end cell row cell k open parentheses 9 t squared minus t to the power of 4 close parentheses end cell cell 0 less or equal than t less or equal than 2 end cell row 1 cell t less or equal than 2 end cell end table close

Show that k equals 1 over 20 .

5b
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2 marks

Find the probability that a randomly selected child takes longer than 90 minutes to tidy their room.

5c
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3 marks

Find the probability density function straight f left parenthesis t right parenthesis of T.

5d
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3 marks

Find the mean time taken for a child to tidy their room.

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6a
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3 marks

The mass, in milligrams, of an ant 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 fraction numerator m minus 1 over denominator k end fraction end cell cell 1 less or equal than m less than 3 end cell row cell 1 over k end cell cell 3 less or equal than m less than 5 end cell row 0 cell otherwise. end cell end table close

Show that k equals 4 .

6b
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6 marks

Fully define the cumulative distribution function straight F left parenthesis m right parenthesis of M.

6c
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2 marks

Find the probability that a randomly selected ant weighs between 2 and 4 milligrams.

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1a
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4 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

 straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 2 end cell row cell a minus b over x end cell cell 2 less or equal than x less than 4 end cell row cell fraction numerator x squared plus 17 over denominator c end fraction end cell cell 4 less or equal than x less than 7 end cell row 1 cell x greater or equal than 7 end cell end table close

(i)
Explain why b equals 2 a.
(ii)
Show that c equals 66.
(iii)
Show that 4 a minus b equals 2 and hence find the values of a and b.
1b
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1 mark

Write down the median of X.

1c
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3 marks

Find the lower quartile and the upper quartile of X.

1d
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2 marks

Describe the skewness of X. Justify your answer.

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2a
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3 marks

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

straight f left parenthesis x right parenthesis equals open curly brackets table row cell x over 5 end cell cell 0 less than x less or equal than 2 end cell row cell fraction numerator 10 minus 2 x over denominator 15 end fraction end cell cell 2 less than x less or equal than 5 end cell row 0 cell otherwise. end cell end table close

Sketch the probability density function of X.

2b
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1 mark

Write down the mode of X.

2c
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7 marks

Fully define the cumulative distribution function straight F open parentheses x close parentheses of X.

2d
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3 marks

Find the median of X.

2e
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2 marks

Comment on the skewness of the distribution of X. Justify your answer.

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3a
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3 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than a end cell row cell fraction numerator 2 x plus 3 over denominator 11 end fraction end cell cell a less or equal than x less or equal than b end cell row 1 cell x greater than b end cell end table close

Show that X follows a continuous uniform distribution on the interval open square brackets a comma b close square brackets and state the values of a and b.

3b
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3 marks

Find the mean and the standard deviation of X.

3c
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2 marks

Find the interquartile range of X.

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4a
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4 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 3 end cell row cell a x squared plus b x end cell cell 3 less or equal than x less or equal than 5 end cell row 1 cell x greater than 5 end cell end table close

Show that a equals 1 over 10 and find the value of b.

4b
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3 marks

Fully define the probability density function straight f open parentheses x close parentheses.

4c
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1 mark

Write down the mode of X.

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5a
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2 marks

Every morning, Simon jogs for an hour and records his distance, D km. D can be modelled as a continuous random variable with a cumulative distribution function straight F open parentheses d close parentheses given by

F open parentheses d close parentheses equals open curly brackets table row 0 cell d less than 8 end cell row cell k open parentheses d minus 8 close parentheses cubed end cell cell 8 less or equal than d less than 9 end cell row cell fraction numerator 55 d squared minus 3600 over denominator 19 d squared end fraction end cell cell 9 less or equal than d less than 10 end cell row 1 cell d greater or equal than 10 end cell end table close

where k is a constant.

Find the value of k.

5b
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2 marks

Find the probability that on a random jog, Simon runs 9.5 km in less than an hour.

5c
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5 marks

Specify fully the probability density function straight f open parentheses d close parentheses of D.

5d
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5 marks

Find the mean distance travelled by Simon on a one-hour jog.

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6a
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3 marks

Mrs Wiltshire teaches college students on the Advanced Integration course. Students are given a percentage score, S percent sign,  at the end of the course. The score S can be modelled using a continuous random variable with cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses s close parentheses equals open curly brackets table row 0 cell s less than 0 end cell row cell 3 over 10000 s squared minus 1 over 500000 s cubed end cell cell 0 less or equal than s less or equal than 100 end cell row 1 cell s greater than 100 end cell end table close

Show that for 0 less or equal than s less or equal than 100, the probability density function can be written in the form

straight f left parenthesis s right parenthesis equals 3 over 500000 space s left parenthesis k minus s right parenthesis

where k is an integer to be found.

6b
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1 mark

Hence write down the mean score.

6c
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4 marks

Find the standard deviation of the scores.

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1a
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1 mark

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 0 end cell row cell k open parentheses 300 x plus 3 x squared minus x cubed minus x to the power of 4 close parentheses end cell cell 0 less or equal than x less or equal than 4 end cell row 1 cell x greater than 4 end cell end table close

Show that k equals 1 over 928.

1b
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3 marks

Show that the median of X is 1.55 correct to 3 significant figures.

1c
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7 marks

Find the mode of X. Fully justify your answer.

1d
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3 marks

Find straight E left parenthesis X right parenthesis.

1e
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2 marks

Comment on the skewness of the distribution of X. Give a reason for your answer.

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2a
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4 marks

The continuous random variable, X, has a cumulative distribution function straight F open parentheses x close parentheses  given by

straight F open parentheses x close parentheses equals open curly brackets table row 0 cell x less than 1 end cell row cell a x squared plus c end cell cell 1 less or equal than x less than 2 end cell row cell b x plus c end cell cell 2 less or equal than x less than 5 end cell row 1 cell x greater or equal than 5 end cell end table close

Find the values of a comma space b and c.

2b
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2 marks

Find the median of X.

2c
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5 marks

Find the lower quartile and the upper quartile of X.

2d
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2 marks

Describe the skewness of X. Give a reason for your answer.

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3a
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3 marks

Constance is a motivational speaker who delivers talks to employees at different companies to promote teamwork, her talks last for at least k minutes. The length, in minutes, of a talk made by Constance is modelled by the continuous random variable, L, which has a cumulative distribution function given by

straight F open parentheses L close parentheses equals open curly brackets table row 0 cell l less than k end cell row cell 1 over 88 open parentheses l squared minus l minus a close parentheses end cell cell k less or equal than l less or equal than 10 end cell row 1 cell l greater than 10 end cell end table close

where a is a constant.

Find the values of a and k.

3b
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4 marks

Find the mean length of time of Constance’s talks.

3c
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2 marks

A company asks Constance to deliver a talk. Find the probability that the talk will last longer than the mean length of time of Constance’s talks.

3d
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2 marks

Without further calculations, state whether the median length of time is smaller than, bigger than or equal to the mean length of time. Give a reason for your answer.

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4
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7 marks

The continuous random variable, Y, has probability density function g open parentheses x close parentheses given by

g open parentheses y close parentheses equals open curly brackets table row cell 1 over 8 y to the power of n end cell cell 0 less or equal than y less or equal than 2 end cell row cell fraction numerator a over denominator 5 y cubed end fraction end cell cell y greater than 2 end cell row 0 cell otherwise. end cell end table close

where a and n are constants.

The cumulative distribution function G open parentheses y close parentheses is given by

G open parentheses y close parentheses equals open curly brackets table row 0 cell y less than 0 end cell row cell r y to the power of 5 end cell cell 0 less or equal than y less or equal than 2 end cell row cell s minus t over y to the power of m end cell cell y greater than 2 end cell end table close

where r comma space s comma space t and m are constants.

Find the values of a comma m comma n comma r comma s space and space t.

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5a
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7 marks

Glen attends the annual Soaring Sunflowers event where thousands of people gather and compare the sizes of their sunflowers. The continuous random variable H represents the heights, in metres, of the sunflowers at the event. The probability density function of H is given by

straight f open parentheses h close parentheses equals open curly brackets table row cell fraction numerator 16 h minus 4 h squared minus 15 over denominator 2 end fraction end cell cell 1.5 less or equal than h less than 2 end cell row cell fraction numerator 5 open parentheses h minus 2 close parentheses over denominator 3 end fraction end cell cell 2 less or equal than h less than 3 end cell row 0 cell otherwise. end cell end table close

Specify fully the cumulative distribution function straight F open parentheses h close parentheses  of H.

5b
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4 marks

The height of Glen’s sunflower is in the top 10% of all heights.

Find an estimate for the minimum height of Glen’s sunflower. Give your answer to 2 decimal places.

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6a
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8 marks

An amusement park has a challenge where guests attempt to remain on a mechanical bull for as long as possible. The times, in seconds, that the guests stay on the mechanical bull can be modelled using a continuous random variable, T, with probability density function straight f open parentheses t close parentheses.

straight f open parentheses t close parentheses equals open curly brackets table row 0 cell t less than 0 end cell row cell 3 over 50 end cell cell 0 less or equal than t less than 5 end cell row cell fraction numerator 6 t minus 15 over denominator 250 end fraction end cell cell 5 less or equal than t less than 10 end cell row cell 1 over t squared end cell cell t greater or equal than 10 end cell end table close

Specify fully the cumulative distribution function straight F open parentheses t close parentheses .

6b
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4 marks

Find the median length of time that a guest is able to stay on the mechanical bull.

6c
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3 marks

Given that a guest has remained on the mechanical bull for 5 seconds, find the probability that the guest will still be on the mechanical bull after another 5 seconds. 

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