1.   [15 marks]Answe_r this question in a separate book

a)  [7  marks]   Let A and B be two events in some sample space, with P(A) > 0 and P(B) > 0.

i)  [3 marks]  Show that, if A and B are independent, they cannot be mutually exclusive.

ii)  [2 marks] Show that, if A and Bare mutually exclusive, they cannot be independent.

iii)  [2 marks]  Suppose now that A and B are mutually exclusive.

Show that

P(AJA U B) = P(A) + P(B) 

b)   [8 marks]

Let X and Y be random variables with joint density function

fx,Y = xy,       Q < X < 2,        0<y<L

i)  [2 marks]  Determine the marginal density fx(x).

ii)  [4 marks]   Set U = 2X and V = Y2 .   Determine the joint density function fu,v(u, v).

iii)  [2 marks]  Compute P(U  是' V  ½)-

2.   [15 marks]Answer this question in a separate book                         

a)  [6  marks]   Let X  ~ N(O, 1) and Y ~ N(O, 1) be two independent standard normal random variables and set W = X + Y.

i)  [4 marks] Using the moment generating functions mx(u) and my(u), determine the moment generating function mw(u) .

ii)  [2 marks]  Hence, or otherwise, deduce that

W ~ N(O, 2).

b)  [9 marks]  Let X1, X2, ... , Xn be a random sample from a random vari­ able X with JE(X) =µand Var(X) = CT2   < oo. Let Xn = ¾ I::t=1 Xi.

i)  Using properties of expectations, show that JE(Xn) = µ .

ii)  Using properties of expectations, show that Var(Xn ) = 号.

iii)  Use Chebychev's inequality to prove that Xn is a consistent estimator of µ, that is, to show that

3.   [15 marks]Answer this question in a separate book

a)   [7 marks]

i)   [2 marks]  State the Central Limit theorem in one of the forms given

in lectures.

ii)   [5  marks]   Let X1, X2, ... , X48  be i.i.d Uniform(0,1) random vari­ ables, and set

-      1   i=48

Using the Central Limit Theorem, compute P(X > 0.55).

Express your answer in terms of the <p function.

In your answer include a sketch of a suitable curve with a shaded area corresponding to this probability.

b)   [8 marks]  A chewing gum manufacturer claims that the gum is produced with average thickness of 7.5 one-hundredths of an inch.   This claim is routinely checked by randomly checking 10 pieces of gum from each production run and measuring their thicknesses.

One such sample of thicknesses is

{7.65, 7.60, 7.65, 7.70, 7.55, 7.55, 7.4, 7.4, 7.5, 7.5},

with sample mean of 7.5500 and a sample standard deviation of 0.1027. Use the graphs and quantiles on the next page when answering this ques­ tion.

i)   [3 marks]  Compute a 90% confidence interval for the mean thickness of the chewing gum.

ii)   [3 marks]  Explain what assumption or assumptions you need to make to compute the confidence interval. If possible, explain whether your assumptions are plausible.

iii)   [2 marks]   Explain in a sentence what the manufacturer can con­                          

clude about whether the average thickness of gum is 7.5 one-hundredths of an inch.