Reference no: EM132524015

SECTION A

Answer all four parts of question 1.

Question 1. (a) For each one of the statements below say whether the statement is true or false, explaining your answer. Throughout this question A and B are events such that 0 < P (A) < 1 and 0 < P (B) < 1.
i. If A and B are independent, then P (A) + P (B) > P (A ∪ B).
ii. If P (A|B) = P (A|B^c) then A and B are independent.
iii. If A and B are disjoint events, then A^c and B^c are disjoint.
iv. Let S and T be unbiased estimators of the parameter θ. Then ST is an unbiased estimator of θ².
v. The power of a test is the probability that the alternative hypothesis is false.

(b) The random variable X has density function given by f (x) = kx²(1-x) for 0 < x < 1 (and 0 otherwise), where k > 0 is a constant.
i. Find the value of k.
ii. Compute Var(1/X).

(c) Urns A, B and C contain, respectively: A: 1 white, 2 black and 3 red balls, B: 2 white, 1 black and 2 red balls, C: 4 white, 5 black and 3 red balls.
i. An urn is chosen at random and two balls are drawn (without replacement) which are white and red. What is the probability that they came from urn B or C?
ii. Suppose two balls are drawn at random (without replacement) from urn C. What is the probability that the balls have the same colour?

(d) The amount of coffee dispensed into a coffee cup by a coffee machine follows a normal distribution with mean 150 ml and standard deviation 10 ml. The coffee is sold at the price of £1 per cup. However, the coffee cups are marked at the 137 ml level, and any cup with coffee below this level will be given away free of charge. The amounts of coffee dispensed in different cups are independent of each other.
i. Find the probability that the total amount of coffee in 5 cups exceeds 700 ml.
ii. Find the probability that the difference in the amounts of coffee in 2 cups is smaller than 20 ml.
iii. Find the probability that one cup is filled below the level of 137 ml.
iv. Find the expected income from selling one cup of coffee.

SECTION B

Answer all three questions from this section.

Question 2. (a) Consider two random variables X and Y taking the values 0 and 1. The joint probabilities for the pair are given by the following table.

	X = 0	X = 1
Y = 0	1 - 3α	α
Y = 1	α	α

i. What are values α can take? Justify your answer.

Now let α = 1/6, and

U = |X - Y |/2 and V = (X + Y - XY)/3.

ii. Compute the mean of U and the mean of V.
iii. Are U and V independent? Justify your answer.

(b) Suppose that you are given observations y₁, y₂ and y₃ such that:
y₁ = 2α + β + ε₁ y₂ = -α - β + ε₂ y₃ = -α + β + ε₃.

The random variables εi, for i = 1, 2, 3, are independent and normally distributed with mean 0 and variance σ₂, while α and β are unknown parameters. Find the least squares estimator of β, verify that it is an unbiased estimator of β, and find the variance of this estimator.

Question 3. (a) The audience shares (in %) of three major television networks' evening news broadcasts in four major cities were examined. The average audience share for the three networks (A, B and C) were 21.35%, 17.28% and 20.18%, respectively. The following is the calculated ANOVA table with some entries missing.

Source	Degrees of freedom	Sum of squares	Mean square	F -value
City Network Error			1.95
Total		51.52

i. Complete the table using the information provided above.
ii. Test, at the 5% significance level, whether there is evidence of a difference in audience shares between networks.

(b) A random sample of size 8 yielded x1 = 1, x2 = 0, x3 = 1, x4 = 1, x5 = 0, x6 = 1, x7 = 1 and x8 = 0. The sample was obtained from the probability function:

p(x; π) = πx (1 - π)1-x for x = 0, 1
                 0 otherwise.

Derive, using the log-likelihood function, the maximum likelihood estimate of the parameter π.

Question 4. (a) Let {X1, . . . , Xn} be a random sample from the (continuous) uniform distribution such that X ∼ Uniform[0, θ], where θ > 0.
i. Find the method of moments estimator (MME) of θ.
Hint: you will need to derive the required population moments.

ii. If n = 3, with the observed data x1 = 0.2, x2 = 3.6 and x3 = 1.1, use the MME obtained in i. to compute the point estimate of θ for this sample. Do you trust this estimate? Justify your answer.
Hint: You may wish to make reference to the law of large numbers.

(b) Let X1, . . . , Xn be i.i.d. copies of a random variable with mean µ and variance
σ2. The method of moments estimator of σ2 is:

σ^2 = 1/n Σ_i=1ⁿ (X - X¯ )² = 1/n Σ_i=1ⁿX_i² - X¯ ².

i. Derive the bias of σ². State whether σ² is a positively or negatively biased estimator of σ².

Hint: You may use without proof the following results: E(X¯ ) = µ and Var(X¯ ) = σ²/n.

ii. Explain how σ² can be transformed into an unbiased estimator of σ².

Report the new estimator.

Attachment:- Statistics.rar