Question

1 Let X be a discrete random variable. (a) Show that if X has a finite mean μ. then EX-ix-0. (b) Show that if X has a finite variance, then its mean is necessarily finite 2 Let X and Y be random variables with finite mean. Show that, if X and Y are independent, then 3 Let Y have mean μ and finite variance σ2 (a) Use calculus to show that μ is the best predictor of Y under quadratic loss. (b) Now suppose that Y has a continuous distribution. Use calculus to show that the median is the best predictor of Y under absolute loss.

Answer 1

1.

X is a discrete random variable

a)

$\mu_x$ is finite mean of X

now

$E[X-\mu_x]=E(X)-E(\mu_x)=\mu_x-\mu_x=0$

Hence Proved

b)

given Var(X) is finite so

$Var(X)<\infty$

this gives

$E(X^2)-(E(X))^2<\infty$ (k)

since Var(X) is finite so

$E(X^2)<\infty$ (l)

from equation(k) and (l)

gives

$(E(X))^2<\infty$

Hence E(X) is finite

2.

if X and Y are independent so

f(x,y) =f(x)*f(y)

so

$E(XY)=\int \int xy f(x,y)dxdy =\int \int xy f(x)f(y)dxdy=\int xf(x)dx*\int yf(y)dy$

$=\mu_x*\mu_y$