Need in an hour

WORKSHEET 1

1. X ~ ( 5, 36) and Y = 0.7X + 2. Compute E[Y] and E[Y2]. Use this to compute Var[Y]. How are the variances of X and Y related?

2. X is a discrete random variable with 0 3 / 4 1 1/ 4

with probability X

with probability 

=  

Show

that E[Xr] = ¼ for all positive integers r.

3. X and Y have a multivariate discrete distribution with the probability function

f(X, Y) where f(-1, 1) = f(0, 0) = f(1, 1) = 1/3. What is the correlation coefficient between X and Y? Compute the marginal density, fx(1) of X when it equals 1. Similarly compute fy(1). Does f(1,1) = fx(1)fy(1) ? Use this example to outline the relationship between correlation and independence.

4. If Y = aX + b, prove ( )sgn *1XY aρ = .

5. Let Y = 0.66X + ε where X and ε are random variables with E[ ε] = 0. What

assumption is necessary for 0.66E Y X x x = =  ?

6. Use the Data Analysis module in EXCEL Tools to draw 100 samples of size 10 from a distribution that is uniform between 0 and 12. Find the mean of the samples and standardise these using the population parameters. Group the standardised means into categories of one standard deviation from zero and compare the histogram of this to that of the standard normal distribution. Repeat the procedure but this time use samples of size 100. Compare and contrast the resulting histograms.