Continuous random variables

A random variable X has probability density function

Use algebraic integration to show that E(X²) =

find, to 3 decimal places, the standard deviation of X

describe, giving a reason, the skewness of the distribution.

Find the cumulative distribution function F(x) for all values of x

Find P(2 X 5)

A random variable X has probability density function

where k and b are constants. Given that E(X í VKRZ WKDW b = 3

The length of time, T, minutes, spent completing a particular task has probability density function

Use algebraic integration to find E(T)

find Var(T)

Find the cumulative distribution function F(t)

Find the 20th percentile of the time taken to complete the task.

Find the probability that the time spent completing the task is more than 1.5 minutes. Given that a person has already spent 1.5 minutes on the task,

find the probability that this person takes more than 3 minutes to complete the task.

The continuous random variable X has the following cumulative distribution function

where k, a and b are constants. Given that the mode of X is

show that a = 4

Find P(X 2.5) giving your answer to 3 significant figures.

The lifetime, X, in tens of hours, of a battery is modelled by the probability density function

Use algebraic integration to find

E(X)

P(X 2.5)

A radio runs using 2 of these batteries, both of which must be working. Two fully-charged batteries are put into the radio.

Find the probability that the radio will be working after 25 hours of use.

Given that the radio is working after 16 hours of use,

find the probability that the radio will be working after being used for another 9 hours.

The continuous random variable X has a probability density function

where k is a positive constant.

Sketch the graph of f (x).

Show that the value of k is

Define fully the cumulative distribution function F(x).

Hence find the 90th percentile of the distribution.

Find P[E(X) X 5.5]

A continuous random variable X has cumulative distribution function F(x) given by

Given that the mode of X is

show that b = 8

find the value of k.

The weight, X kg, of staples in a bin full of paper has probability density function

Use integration to find

E(X)

Var (X)

P(X 1.5)

Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for £50 but if the weight of the staples exceeds 1.5 kg it sells for £25

Find the expected amount of money Peter raises per bin full of paper.

Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have 20% fewer bins full of paper to sell.

Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.

A random variable X has probability density function given by

where k is a constant.

Write down the mode of X.

Specify fully the cumulative distribution function F(x).

Find the upper quartile of X.

The continuous random variable L represents the error, in metres, made when a machine cuts poles to a target length. The distribution of L is a continuous uniform distribution over the interval [0, 0.5]

Find P(L < 0.4).

Write down E(L).

Calculate Var(L).

A random sample of 30 poles cut by this machine is taken.

Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length.

When a new machine cuts poles to a target length, the error, X metres, is modelled by the cumulative distribution function F(x) where

Using this model, find P(X 0.4)

A random sample of 100 poles cut by this new machine is taken.

Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres.

A continuous random variable X has probability density function f(x) where

where k and n are positive integers.

Find k in terms of n.

Find E(X) in terms of n.

Find E(X²) in terms of n.

Given that n = 2

find Var(3 X).

The random variable X has probability density function f(x) given by

where k is a constant.

Sketch f (x).

Write down the mode of X.

describe, giving a reason, the skewness of the distribution.

Use integration to find the value of k.

Write down the lower quartile of X.

find the exact value of P(X 3).

In an experiment some children were asked to estimate the position of the centre of a circle. The random variable D represents the distance, in centimetres, between the child’s estimate and the actual position of the centre of the circle. The cumulative distribution function of D is given by

Find the median of D.

Find the mode of D. Justify your answer.

The experiment is conducted on 80 children.

Find the expected number of children whose estimate is less than 1 cm from the actual centre of the circle.

The continuous random variable Y has cumulative distribution function

where k is a constant.

Find the value of k.

Find the probability density function of Y, specifying it for all values of y.

Find P(Y > 1).

The random variable X has probability density function f(x) given by

where k is a constant.

Find the mode of X.

Use algebraic integration to find E(X).

By comparing your answers to parts (b) and (c),

describe the skewness of X, giving a reason for your answer.

The continuous random variable X has a cumulative distribution function

where a and b are constants.

Find the value of a and the value of b.

Use integration to find E(X).

Show that the lower quartile of X lies between 1.425 and 1.435

The continuous random variable T is used to model the number of days, t, a mosquito survives after hatching. The probability that the mosquito survives for more than t days is

Show that the cumulative distribution function of T is given by

Find the probability that a randomly selected mosquito will die within 3 days of hatching.

Given that a mosquito survives for 3 days, find the probability that it will survive for at least 5 more days.

A large number of mosquitoes hatch on the same day.

Find the number of days after which only 10% of these mosquitoes are expected to survive.

The continuous random variable X has the following probability density function

where a and b are constants.

find a second equation in a and b,

hence find the value of a and the value of b.

Find, to 3 significant figures, the median of X.

Comment on the skewness. Give a reason for your answer.

Figure 1 shows a sketch of the probability density function f (x) of the random variable X. For 0 x 3, f (x) is represented by a curve OB with equation f (x) = kx², where k is a constant. For 3 x a, where a is a constant, f (x) is represented by a straight line passing through B and the point (a, 0). For all other values of x, f (x) = 0. Given that the mode of X = the median of X, find

the mode,

the value of k,

the value of a.

Without calculating E(X) and with reference to the skewness of the distribution

state, giving your reason, whether E(X) 3, E(X) 3 or E(X) 3.

The continuous random variable X has probability density function given by

Sketch f (x) showing clearly the points where it meets the x-axis.

Write down the value of the mean, of X.

The cumulative distribution function of X is given by

where a is a constant.

Find the value of a.

Show that the lower quartile of X, q₁, lies between 2.29 and 2.31

Hence find the upper quartile of X, giving your answer to 1 decimal place.

Find, to 2 decimal places, the value of k so that

A continuous random variable X has the probability density function f(x) shown in Figure 1.

Show that f(x) = 4 – 8x for 0 x 0.5 and specify f(x) for all real values of x.

Find the cumulative distribution function F(x).

Find the median of X.

Write down the mode of X.

State, with a reason, the skewness of X.

The queuing time in minutes, X, of a customer at a post office is modelled by the probability density function

Using integration, find

the mean queuing time of a customer,

the probability that a customer will queue for more than 5 minutes.

Three independent customers shop at the post office.

Find the probability that at least 2 of the customers queue for more than 5 minutes.

The lifetime, X, in tens of hours, of a battery has a cumulative distribution function F(x) given by

Find the median of X, giving your answer to 3 significant figures.

Find, in full, the probability density function of the random variable X.

Find P(X ≥ 1.2)

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.

Find the probability that the lantern will still be working after 12 hours.

The random variable Y has probability density function f(y) given by

where k and a are positive constants.

Given that E(Y) = 1.75

show that a = 4 and write down the value of k.

For these values of a and k,

sketch the probability density function,

write down the mode of Y.

A continuous random variable X has cumulative distribution function

Find P(X < 0).

Find the probability density function f(x) of X.

Write down the name of the distribution of X.

Find the mean and the variance of X.

Write down the value of P(X = 1).

The continuous random variable X has probability density function f(x) given by

where k is a constant.

Find the cumulative distribution function F(x).

Find the mean of X.

Show that the median of X lies between x = 2.6 and x = 2.7

The three independent random variables A, B and C each has a continuous uniform distribution over the interval [0, 5].

Find P(A > 3).

Find the probability that A, B and C are all greater than 3.

The random variable Y represents the maximum value of A, B and C. The cumulative distribution function of Y is

Find the probability density function of Y.

Sketch the probability density function of Y.

Write down the mode of Y.

Find E(Y).

Find P(Y > 3).

Figure 1 shows a sketch of the probability density function f(x) of the random variable X. The part of the sketch from x = 0 to x = 4 consists of an isosceles triangle with maximum at (2, 0.5).

Write down E(X).

The probability density function f(x) can be written in the following form.

Find the values of the constants a and b.

Show that , the standard deviation of X, is 0.816 to 3 decimal places.

Find the lower quartile of X.

State, giving a reason, whether P(2 – X is more or less than 0.5

The length of a telephone call made to a company is denoted by the continuous random variable T. It is modelled by the probability density function

Show that the value of k

Find P(T > 6).

Calculate an exact value for E(T) and for Var(T).

Write down the mode of the distribution of T.

It is suggested that the probability density function, f(t), is not a good model for T.

Sketch the graph of a more suitable probability density function for T.

A random variable X has probability density function given by

Show that the cumulative distribution function F(x) can be written in the form ax² + bx + c, for 1 x 4 where a, b and c are constants.

Define fully the cumulative distribution function F(x).

Show that the upper quartile of X is 2.5 and find the lower quartile.

Given that the median of X is 1.88

describe the skewness of the distribution. Give a reason for your answer.