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L1079 Mathematical Statistics

Practice exercises 2

1. The joint distribution of random variables X and Y is represented in the table below, where the cell values denote joint probabilities.

(a) Obtain the marginal distribution of X.

(b) Find P(X =Y)

(c) Find P(X ≥Y | Y  < 3).

(d) Let Z=X+Y. Obtain the probability distribution of Z.

2. Consider a random sample X1, ...., Xn from a distribution with pdf

f(x) = 0.5(1 + θx), -1 ≤ x ≤ 1

= 0 otherwise

For what value of c is the statistic cX an unbiased estimator of θ?

3. A fair coin is being tossed. Each time it comes up heads, you get £2. Each time it comes up tails you get £1. How much money do you expect to make in 10 tosses of the coin?

4. Suppose that X1, ...., Xn is a random sample drawn from a uniform distribution over the interval (0,θ). Is the sample mean an unbiased estimator of θ?  Be sure to explain your answer clearly.  

5. The length of time required by students to complete a one-hour exam is a random variable with a density function given by

f(y) = cy2 + y, 0 ≤ y ≤ 1

= 0 elsewhere

a. Find c.

b. Find the cdf, F(y).

c. Find the probability that a randomly selected student will finish in less than half an hour, i.e. P(Y<0.5).

d. Given that a particular student needs at least 15 minutes to complete the exam, find the probability that she will require at least 30 minutes to finish.

6. A random variable X has pdf given by:

f(x) = 2e-2x, for x >0

= 0 otherwise

Find the median of X.

7. A poll is conducted to determine the proportion of voters, p, who approve of a presidential candidate. Suppose that 10 voters are polled. We want to test the hypothesis that p=0.5. Suppose our decision rule is to reject the null hypothesis whenever the number of sampled voters who approve of the candidate is fewer than 2.

(i) What is the probability of Type I error associated with our rejection rule?

(ii) What would be the probability of Type II error if in fact p=0.3?                 

8. The exponential density function is given by

f(x) = θe-θx, for x >0

= 0 otherwise

where θ is positive. Given that the mean of this distribution is 1/θ, obtain a Method of Moments estimator of the median, assuming you have a random sample X1, ...., Xn from this distribution.