Continuous uniform distribution

The continuous random variable X is uniformly distributed over the interval ≤ x where and are constants. The mean of X is and the standard deviation is

find the value of and the value of

calculate the exact value of P( X + )

Susie has 30 m of fencing. She wishes to form a rectangular enclosure of perimeter 30 m. The length, Y metres, of one side of the enclosure is uniformly distributed between 4 m and 12 m.

Find the probability that the length of the longer side of the enclosure is more than 8 m.

David aims to catch the train to work each morning. The scheduled departure time of the train is 0830 The number of minutes after 0830 that the train departs may be modelled by the random variable X. Given that X has a continuous uniform distribution over [ α, β ] and that E(X) = 4 and Var(X) = 12

find the value of α and the value of β. Each morning, the probability that David oversleeps is 0.05 If David oversleeps he will be late for work. If he does not oversleep he will be in time to catch the train, but will be late for work if the train departs after 0835

Find the probability that David will be late for work. Given that David is late for work,

find the probability that he overslept.

The continuous random variable X is uniformly distributed over the interval [, ]

find the value of and the value of

find the value of c

find P(c X 9)

A rectangle has a perimeter of 200 cm. The length, S cm, of one side of this rectangle is uniformly distributed between 30 cm and 80 cm.

Find the probability that the length of the shorter side of the rectangle is less than 45 cm.

The random variable R has a continuous uniform distribution over the interval [5, 9]

Specify fully the probability density function of R.

Find P(7 R 10)

The random variable A is the area of a circle radius R cm.

Find E(A)

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 piece of string AB has length 9 cm. The string is cut at random at a point P and the random variable X represents the length of the piece of string AP.

Write down the distribution of X.

Find the probability that the length of the piece of string AP is more than 6 cm.

The two pieces of string AP and PB are used to form two sides of a rectangle. The random variable R represents the area of the rectangle.

Show that R = aX² + bX and state the values of the constants a and b.

Find E(R).

Find the probability that R is more than twice the area of a square whose side has the length of the piece of string AP.

The random variable X has a continuous uniform distribution on [ a, b ] where a and b are positive numbers. Given that E(X) = 23 and Var(X) = 75

find the value of a and the value of b.

Given that P(X > c) = 0.32

find P(23 < X < c).

A continuous random variable X is uniformly distributed over the interval [ b, 4 b ] where b is a constant.

Write down E(X).

Use integration to show that Var(X) =

Find Var(3 – 2 X).

Given that b = 1 find

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

the median of X.

In a village shop the customers must join a queue to pay. The number of customers joining the queue in a 10 minute interval is modelled by a Poisson distribution with mean 3 Find the probability that

exactly 4 customers join the queue in the next 10 minutes,

more than 10 customers join the queue in the next 20 minutes.

When a customer reaches the front of the queue the customer pays the assistant. The time each customer takes paying the assistant, T minutes, has a continuous uniform distribution over the interval [0, 5]. The random variable T is independent of the number of people joining the queue.

Find P(T > 3.5)

In a random sample of 5 customers, the random variable C represents the number of customers who took more than 3.5 minutes paying the assistant.

Find P(C 3)

Bethan has just reached the front of the queue and starts paying the assistant.

Find the probability that in the next 4 minutes Bethan finishes paying the assistant and no other customers join the queue.

The continuous random variable X is uniformly distributed over the interval [4, 6].

Write down the mean of X.

Find P(3 X 5 3)

The continuous random variable Y is uniformly distributed over the interval [ a, 4 a ].

Use integration to show that E(Y²) = 7a²

Find Var(Y).

Given that P(X = P(Y find the value of a.

In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm, of a randomly chosen stick has a continuous uniform distribution over the interval [7, 10]. A stick is selected at random from the box.

Find the probability that the stick is shorter than 9.5 cm.

To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm.

Find the probability of winning a bag of sweets.

To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm.

Find the probability of winning a soft toy.

The continuous random variable X is uniformly distributed over the interval [–1,3]. Find

E(X)

Var(X)

P(X 1.4)

A total of 40 observations of X are made.

Find the probability that at least 10 of these observations are negative.

A rectangle has a perimeter of 20 cm. The length, X cm, of one side of this rectangle is uniformly distributed between 1 cm and 7 cm. Find the probability that the length of the longer side of the rectangle is more than 6 cm long.

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 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).

The continuous random variable X is uniformly distributed over the interval [–2, 7].

Write down fully the probability density function f(x) of X.

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

Find

P(– 0.2 < X < 0.6).