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1 Let X be a discrete random variable with pmf: 0

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1 Let X be a discrete random variable with pmf:

0.3 ifx=1

0.5 ifx=2

T)=

KO) =) > ite a3

QO otherwise.

Suppose Y is a continuous random variable and that conditional on X

it has pdf:

2y

— if0<y<ez

fy|x(ylz) = ¢ x?

0 ify<0Oory>2

(remember that this conditional pdf is only defined for X = 1, 2,3)

 

a Find the joint pdf of X and Y, fx y(z,y). (5 marks)

(Hint: Define the joint pdf separately for each of the three values of X for which fx(x) > 0. In each case remember to also include the range of values of y for which the result holds.)

 

b Find fy(y) the marginal pdf of Y. Remember to include the range of values of y for which the result holds. (4 marks)

 

2 Let X and Y be random variables with joint pdf:

1

—(l+2z if -l<a<l,-l<y<l

fxy (x,y) = a\ ¥) "

0 otherwise

 

a Find the marginal pdf of X, fx(x). Remember to include the range of values of x for which the result holds. (2 marks)

 

b Explain why Y has the same distribution as X. (1 mark)

 

c Find E(X) and Var(X) and hence state E(Y) and Var(Y)

 

(4 marks)

 

d Find E(XY) (2 marks)

 

e Find Cov(X, Y) (2 marks)

 

f Find pxy (2 marks)

 

g Find fy;x(y|x). Remember to state for what values of x the distribution is defined as well as the range of values of y.

 

(4 marks)

h Find E(Y |X) (2 marks)

i Use your answer to 2h to calculate EJ[E(Y|X)] and use this and

your answer to 2c to verify that the partition theorem for expectations (property 6.7) holds in this case. (3 marks)

j Calculate Var(Y|X), and use this to calculate E[Var(Y |X )].

(3 marks)

k Use your answer to 2h to calculate Var[E(Y|X)] (1 mark)

1 Use your answers to 2c, 2j and 2k to show that the variance partition formula (Theorem 6.5) holds in this case. (2 marks)

3 The relation between degrees Celsius(°C') and degrees Fahrenheit(° F’)

1S:

sp 2 0c4 2

5

Let X be the average daily temperatures in degrees Celsius in Dunedin

and let Y be the average daily temperatures in degrees Celsius in

Auckland. Suppose that Cov(X, Y) = 3 and px. y = 0.8. Let T be the average daily temperatures in degrees Fahrenheit in Dunedin and S the average daily temperatures in degrees Fahrenheit in Auckland.

Compute Cov(T, S) and prs. (3 marks)

4 Suppose that X and Y are independent variables both with a Gamma(k = 2, \ = 0.5) distribution.

a Use aconvolution integral to find the pdf of Z = X+Y. Remember to include the range of values of z for which the result holds.

(5 marks)

b Z has a named distribution. State the name of the distribution and give its parameters. Explain why it makes sense that Z has this distribution. (3 marks)

c Find the joint pdf of X and Y, fx y(z,y). Remember to include the range of values of x and y for which the result holds.

(2 marks)

d IfW = X —Y. Use the result of Example 6.12 to find the pdf of Z and W, fzw/(z,w) Remember to include the range of values of z and w for which the result holds. (3 marks)
5 If X and Y are two independent continuous random variables and

Z x th

= — then

Y

oOo

fz(z) = / fx (za) fy(x)|2| dx for — co < z < 00

— Oo

Suppose that X and Y both have a standard Normal distribution. Use the above information to show that

1

Ial2)= ay for —o <z< oO.

Hint:

oO OO

D5 | lnjew 22 +)?" dy = 2 [cto dx

27 27

—oo 0

(2 marks)

Total: 55 marks.