(a) (4 pts) Let X,Y be (not necessarily independent) random variables (with finite mean and
variance). Prove that (E(XY))2E(X2)E(Y2) (Hint: Consider the function g(t)=E((X+Yt)2). Show that
it is a quadratic function. Since g(t)0 for all t, its discriminant is nonpositive.) (b) (4 pts) Let X be a
random variable (with finite mean and variance). Let =E(X), 2=Var(X). The kurtosis of X is defined
as Kurt(X)=4E((X)4) Fix any a,bR,a=0, and let Y=aX+b. Prove that Kurt(Y)=Kurt(X). (c) (4 pts)
Continuing part (b), prove that Kurt(X)1. (Hint: Use part (a) on a suitable pair of random variables.)
1 (d) (4 pts) Continuing part (b), prove that for any >0, Pr(X)4Kurt(X) (Hint: Use Markov's inequality
on a suitable random variable.).
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a 4 pts Let XY be not necessarily independent random .pdf
1. (a) (4 pts) Let X,Y be (not necessarily independent) random variables (with finite mean and
variance). Prove that (E(XY))2E(X2)E(Y2) (Hint: Consider the function g(t)=E((X+Yt)2). Show that
it is a quadratic function. Since g(t)0 for all t, its discriminant is nonpositive.) (b) (4 pts) Let X be a
random variable (with finite mean and variance). Let =E(X), 2=Var(X). The kurtosis of X is defined
as Kurt(X)=4E((X)4) Fix any a,bR,a=0, and let Y=aX+b. Prove that Kurt(Y)=Kurt(X). (c) (4 pts)
Continuing part (b), prove that Kurt(X)1. (Hint: Use part (a) on a suitable pair of random variables.)
1 (d) (4 pts) Continuing part (b), prove that for any >0, Pr(X)4Kurt(X) (Hint: Use Markov's inequality
on a suitable random variable.)
