State whether each of the following is always true (T) or notalways 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 Xand 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 Xand Y is given simply by f(x, y) = fx(x)fy(y|X = x)
d) If X1, X2, ..., Xn are independent random variables, eachfollowing a Bernoulli distribution with the same parameter p, thenthe sum Σn i=1Xi is a Binomial random variables, with parameters nand p.
e)If X1, X2, ..., X100 are independent, normally distributedrandom variables, then the average X¯ = 1 100Σ 100 i=1Xi of theserandom variables is itself a random variable following a normaldistribution.
f) If Z1 and Z2 are independent random variables, each followinga standard normal distribution, then Z1 + Z2 follows a standardnormal distribution as well.
g) If X ∼ χ 2 (4) and Y ∼ t(1) then the 95th percentile of Xexceeds the 95th percentile of Y .
h) If X ∼ F(5, 7) and Y ∼ F(7, 5), then the 5th percentile of Xis greater than the 5th percentile of Y .
