Question

State whether each of the following is always true (T) or not always true (F).

a) If X is a random variable, Corr X, (1/3)X= (1/3).

b) If X and Y are independent random variables then E(X|Y ) = E(X)

c) d) If fx(x) is the marginal density of a random variable X and fy(y|X = x) is the conditional density of a random variable Y , given a particular realization x of X, then the joint density of X and Y is given simply by f(x, y) = fx(x)fy(y|X = x)

d) If X1, X2, ..., Xn are independent random variables, each following a Bernoulli distribution with the same parameter p, then the sum Σn i=1Xi is a Binomial random variables, with parameters n and p.

e)If X1, X2, ..., X100 are independent, normally distributed random variables, then the average X¯ = 1 100Σ 100 i=1Xi of these random variables is itself a random variable following a normal distribution.

f) If Z1 and Z2 are independent random variables, each following a standard normal distribution, then Z1 + Z2 follows a standard normal distribution as well.

g) If X ∼ χ 2 (4) and Y ∼ t(1) then the 95th percentile of X exceeds the 95th percentile of Y .

h) If X ∼ F(5, 7) and Y ∼ F(7, 5), then the 5th percentile of X is greater than the 5th percentile of Y .

Answer 1