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1. The continuous random variable X has probability density function f (x) where

fk(x) =

otherwise,0

10,ee xk kx

(a) Show that k = 1. (3)

(b) What is the probability that the random variable X has a value that lies between

41 and

21 ? Give your answer in terms of e.

(2)

(c) Find the mean and variance of the distribution. Give your answers exactly, in terms of e. (6)

The random variable X above represents the lifetime, in years, of a certain type of battery.

(d) Find the probability that a battery lasts more than six months. (2)

A calculator is fitted with three of these batteries. Each battery fails independently of the other

two. Find the probability that at the end of six months

(e) none of the batteries has failed; (2)

(f) exactly one of the batteries has failed. (2)

(Total 17 marks)

2. The lifetime of a particular component of a solar cell is Y years, where Y is a continuous random

variable with probability density function

0. when e5.0

0 when 0)(

/2- y

yyf

y

(a) Find the probability, correct to four significant figures, that a given component fails

within six months.

Each solar cell has three components which work independently and the cell will continue to run

if at least two of the components continue to work. (3)

(b) Find the probability that a solar cell fails within six months. (4)

(Total 7 marks)

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3. A continuous random variable X has probability density function

elsewhere

,10for,

,0

)1(

4

)(2

xxxf

Find E(X).

(Total 3 marks)

4. The probability density function f(x), of a continuous random variable X is defined by

f(x) =

otherwise.,0

20),–4(4

1 2 xxx

Calculate the median value of X.

(Total 6 marks)

5. The discrete random variable X has the following probability distribution.

P(X = x) =

otherwise,0

4,3,2,1, xx

k

Calculate

(a) the value of the constant k;

(b) E(X).

(Total 6 marks)

6. Let f(x) be the probability density function for a random variable X, where

f(x)

otherwise,0

20for,2 xkx

(a) Show that k = 8

3.

(2)

(b) Calculate

(i) E(X);

(ii) the median of X. (6)

(Total 8 marks) 7. A continuous random variable X has probability density function given by

3

f(x) = k (2x – x2), for 0 x 2

f(x) = 0, elsewhere.

(a) Find the value of k.

(b) Find P(0.25 x 0.5).

(Total 6 marks)

8. The continuous random variable X has probability density function

f(x) = 6

1x(1 + x

2) for 0 x 2,

f(x) = 0 otherwise.

(a) Sketch the graph of f for 0 x 2. (2)

(b) Write down the mode of X. (1)

(c) Find the mean of X. (4)

(d) Find the median of X. (5)

(Total 12 marks) 9. The probability density function f(x) of the continuous random variable X is defined on the

interval [0, a] by

. 3for 8

27

.3 0for 81

)(

2ax

x

xxxf

Find the value of a.

(Total 6 marks)