A continuous random variable has the probability density function given by
Find the value of .
Giving your answers to three significant figures, find
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A continuous random variable has the probability density function given by
Find the value of .
Giving your answers to three significant figures, find
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The continuous random variable has probability density function
Find the value of .
Sketch the probability density function.
Find:
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The discrete random variable, , has probability distribution function
Show that .
Find the expected value and variance of .
As part of a game, a four-sided spinner is created with the numbers 2, 4, 6 and 12. The discrete random variable is used to model the number that the spinner lands on. The score allocated to a player on their turn is 4 more than double the value the spinner lands on.
Find the expected value and variance of a player’s score from a single spin.
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A UK energy company charges £0.22 per kilowatt hour (kWh) of electricity used.
The amount of energy used per day by the company’s customers, kWh, follows the following probability density function
Show that .
A customer’s total daily charge consists of a fixed (standing) charge of £0.38 per day plus the charge for the electricity used.
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Consider the function defined by,
Use your GDC to verify that can represent a probability density function for a continuous random variable in the case when . Explain your verification process in full.
Use your GDC to find the mode of .
Use your GDC to estimate the median of .
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Consider the probability density function for a continuous random variable,
Find
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In a quick-fire quiz consisting of 25 questions, contestants have just over 2 seconds to answer each question. The time taken on any single question in the quiz is modelled by the continuous random variable, , which has probability density function
Find the exact value of and verify that this is just over 2 seconds.
Find the probability that a contestant takes between 1 and 2 seconds to answer a question.
Sketch the graph of
Write down the mode of .
Find the median of .
Find the mean time for a contestant to answer all 25 questions.
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The continuous random variable, , has probability density function
Given that , write down, in terms of , the mean, median and mode of , briefly explaining how you obtained your answers.
and are values of such that .
Write down an equation connecting and .
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Two random variables, and are such that .
It is also known that and .
Show that .
Given that , find .
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Consider the function
Sketch the graph of .
Write down a domain for such that it could be a probability density function for a continuous random variable .
Write down
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A ‘lucky dip’ bag contains seven bars of chocolate and 5 packets of sweets. Suraya selects two items at random without replacing them.
The probability distribution table for the discrete random variable , “the number of packets of sweets selected”, is shown below.
0 |
1 |
2 |
|
Find the value of .
Find
Find .
Find
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A population of grasshoppers is being studied. It is found that the length of an adult grasshopper, in cm, has
Find the value of .
Sketch the probability density function.
Find the probability that a grasshopper picked at random is less than 4 cm in length.
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A game is played with two fair spinners. Each spinner is divided into three sections numbered 1, 2 and 3. A player’s score is obtained by spinning both spinners simultaneously and adding together the numbers that they land on.
Complete the table below for the probability distribution of the game.
Score, | |||||
Find the expected score,
Jian Wei wants to award prizes such that a player receives $3 for the score that they achieve.
Find the expected prize money for the game.
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A continuous random variable has a probability distribution function
Show that the mean of the random variable is equal to 1.
Find the variance of the random variable.
Hence, find the standard deviation of the random variable, leaving your answer in the form .
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At a school probability fair, some students create a game using one complete suit from a standard pack of cards. A player must pay $1 to pick a card at random. If their card is a jack, queen or a king they will receive $1 back, if their card is an ace they will receive $5 otherwise if their card is an ordinary number card from 2 to 10, they will receive nothing.
Show that the game is not fair.
Calculate
The students want to make the game fair, so decide to give a prize to anyone who picks an ordinary number card.
Calculate the value of the new prize for choosing an ordinary number card.
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A discrete random variable has probability distribution given by , where .
Find the value of
Complete the probability distribution table below.
B | 5 | 6 | 7 |
Find the mean of B.
Find the standard deviation of B.
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A continuous random variable has the probability density function given by
Find the value of .
Hence, find the values of
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A random variable has and .
Find
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Consider the function defined by
where is the probability density function of a continuous random variable.
Sketch the graph of
Find the value of .
Find the value of .
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The continuous random variable has the probability density function
where and are constants.
Given that is a continuous function, find the values of and .
Find the median of .
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The continuous random variable, , follows a uniform distribution.
The probability density function is given by
Given that and , find the values of and .
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A company meeting lasts hours. The continuous random variable can be modelled by the probability density function
Find the mean time of a meeting, giving your answer in minutes.
Show that the median meeting time is greater than the modal meeting time. Fully explain your solution.
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The continuous random variable has probability density function
Show that .
Given that show that and hence find the value of .
Find the exact value of .
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Paul is travelling around France to try to find the perfect baguette. The continuous random variable represents the cost, in euros, of a baguette. It can be modelled by the probability density function
Sketch the graph of .
Explain why the mean cost of a baguette is €7.
Given that the probability that a randomly selected baguette costs less than €9 is , find the probability that a randomly selected baguette costs between €5 and €9
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The amount of time, years, it takes for an investment to return a profit is modelled by a continuous random variable, , with probability density function
where and are integer constants.
Given that the mode of is 1, find the values of and .
Find the expected length of time that a typical investor will have to wait before they make a profit. Give your answer in months, to the nearest month.
A particular company owner will only commit to this investment if there is at least a 75% chance they will make a profit within the first 18 months. Determine whether the company owner will invest or not.
Give a criticism of the model.
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The continuous random variable, , has probability density function given by
Draw a box plot for the distribution of . Mark the exact values of the quartiles on your diagram.
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The discrete random variable, , has the probability distribution given in the table below
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
|
where .
Show that .
Find
Find
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A continuous random variable, , has a probability distribution such that
Given that , find the value of .
Find
Find
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The function defined by has a local maximum when and a local minimum when .
Sketch the graph of . State the values of p and q.
The continuous random variable, , has probability density function
Briefly explain how it can be verified that is a suitable model for a probability density function.
Write down the mode of .
Find the probability that
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