MA22S6
Assignment 4 - joint distributions and sampling
Mike Peardon (mjp@maths.tcd.ie)
February 16, 2012
Answer all four parts. Each part is worth five marks. Hand the homework in by the end of next Friday’s tutorial (24th
February).
Question 1
22S6 Travel sells package holidays for students in “reading week”. A survey rates customer satisfaction for the flight and
accommodation separately. The satifaction index for the flight is a random number X in the range [0, 1] and the index for
the accomodation is a random number Y in the same range. The joint probability density for X and Y is
f X ,Y (x, y) = x + y
Are X and Y independent? Find the probability a customer is dissatisfied (dissatisfied means the satisfaction index is less
than 1/2) by
1. the flight alone
2. either the flight or the hotel.
Question 2
Two random numbers X 1 and X 2 are independent and identically distributed with mean µ and variance σ2 . A weighted
mean is constructed X = αX 1 + (1 − α)X 2 . Find expressions for the mean and variance of X and determine the value of α
for which the variance is minimised.
Question 3
A random number A takes values ±1 with equal probability.
1. Find the expected value µ and standard deviation σ A of A
2. Ā is defined as the sample mean of 20 independent copies of A. What is the variance of Ā?
1 1
3. Find P 0 , P 1 , P 2 and P 3 , the probabilities Ā takes values 0, 10
, 5 and
theorem tell you should be happening?
3
10
correspondingly. What does the central limit
Question 4
The five values tabulated below are independent samples of a random number U with unknown expected value µU and
variance 0.4. Use the data to compute the sample mean and hence quote a range of values in which you are 95% confident
that µU must lie. The 95% confidence interval corresponds to two standard deviations.
Table 1
U1 0.153
U2 0.457
U3 0.388
U4 2.752
U5 1.651