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Question 5 [27 marks] a) The joint probability distribution of two discrete random variables X and Y is given below. X = 1 X = 2 X = 3 X = 4 Y = 1 0.06 0.02 0.2 0.18 Y = 2 0.12 0 p 0.06 Y = 3 0 0.08 0.16 0 i. Find the probability p. ii. Find the marginal distributions f_X(x) and f_Y(y). iii. Find the conditional distribution f_{X|Y=2} iv. Find P(X > Y). b) The probability density function of a continuous random variable T is given by f(t) = { kt sin(?t), 0 ? t ? 1; 0, otherwise Find k. c) The cumulative distribution function of a random variable is given by F(x) = { (1/2)exp((x-?)/b), x ? ?; 1 - (1/2)exp(-(x-?)/b), x ? ? Where b is a positive constant and the median of the distribution is ?. Find the upper quartile of this distribution.

          Question 5 [27 marks]

a) The joint probability distribution of two discrete random variables X and Y is given below.

	X = 1	X = 2	X = 3	X = 4
Y = 1	0.06	0.02	0.2	0.18
Y = 2	0.12	0	p	0.06
Y = 3	0	0.08	0.16	0

i. Find the probability p.

ii. Find the marginal distributions f_X(x) and f_Y(y).

iii. Find the conditional distribution f_{X|Y=2}

iv. Find P(X > Y).

b) The probability density function of a continuous random variable T is given by
f(t) = { kt sin(?t), 0 ? t ? 1; 0, otherwise
Find k.

c) The cumulative distribution function of a random variable is given by
F(x) = { (1/2)exp((x-?)/b), x ? ?; 1 - (1/2)exp(-(x-?)/b), x ? ?
Where b is a positive constant and the median of the distribution is ?.
Find the upper quartile of this distribution.

Question 5 [27 marks]

a) The joint probability distribution of two discrete random variables X and Y is given below.

X = 1 X = 2 X = 3 X = 4
Y = 1 0.06 0.02 0.2 0.18
Y = 2 0.12 0 p 0.06
Y = 3 0 0.08 0.16 0

i. Find the probability p.

ii. Find the marginal distributions fX(x) and fY(y).

iii. Find the conditional distribution fX|Y=2

iv. Find P(X > Y).

b) The probability density function of a continuous random variable T is given by
f(t) = kt sin(?t), 0 ? t ? 1; 0, otherwise
Find k.

c) The cumulative distribution function of a random variable is given by
F(x) = (1/2)exp((x-?)/b), x ? ?; 1 - (1/2)exp(-(x-?)/b), x ? ?
Where b is a positive constant and the median of the distribution is ?.
Find the upper quartile of this distribution.

Added by Thomas R.

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