# Stats review

• Oct 15th 2008, 12:17 PM
plm2e
Stats review
hello, I am working on a review sheet to get ready for a stats test. I am not finding any examples in my text book to help me. I have a few questions and I am looking for an explanation, not just the answer. Please help me out if you can.

1. Let X1, X2, ... ,Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum (X1, X2, ... ,Xn). Show that the distribution function of M, FM(.) is given by: FM(x) = x^n , 0< x < 1.

2. The joint density of X and Y is given by:

f(x,y) = [e^(-x/y)e^(-y)]/y , 0 < x < ¥, 0 < y < ¥

Show that E[XlY=y] = y

3. If X and Y are both discrete, show that ∑x Pxly (xly)= 1 for all such that PY(y) > 0.

4. Show that
a. Cov (X,Y) = Cov (X,E[YlX])
b. suppose, that, for constants a and b, E[YlX] = a + bX
show that b= Cov(X,Y)/Var(X)
• Oct 15th 2008, 07:06 PM
mr fantastic
Quote:

Originally Posted by plm2e
hello, I am working on a review sheet to get ready for a stats test. I am not finding any examples in my text book to help me. I have a few questions and I am looking for an explanation, not just the answer. Please help me out if you can.

1. Let X1, X2, ... ,Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum (X1, X2, ... ,Xn). Show that the distribution function of M, FM(.) is given by: FM(x) = x^n , 0< x < 1.

[snip]

$F_M (x) = \Pr(M \leq x) = \Pr(X_1 \leq x, \, X_2 \leq x, \, .... \, X_n \leq x)$ $= \Pr(X_1 \leq x) \cdot \Pr(X_2 \leq x) \cdot .... \Pr(X_n \leq x)$ $= \left[ G(x) \right]^n$

(since the X's are i.i.d) where G(x) is the cdf of X.

And you should know that $G(x) = \int_{0}^{x} 1 \, du = x$ for $0 \leq x \leq 1$.
• Oct 15th 2008, 07:34 PM
plm2e
thanks for the help on prob 1.
• Oct 15th 2008, 07:53 PM
mr fantastic
Quote:

Originally Posted by plm2e
[snip]

2. The joint density of X and Y is given by:

f(x,y) = [e^(-x/y)e^(-y)]/y , 0 < x < ¥, 0 < y < ¥

Show that E[XlY=y] = y

[snip]

From the definition: $E(X \, | \, Y = y) = \int_{0}^{+\infty} x \, f(x \, | \, Y = y) \, dx$.

From the definition: $f(x \, | \, Y = y) = \frac{f(x, y)}{f_Y (y)}$.

From the definition: $f_Y(y) = \int_{0}^{+\infty} f(x, y) \, dx = \int_{0}^{+\infty} \frac{e^{-x/y} \, e^{-y}}{y} \, dx = \frac{e^{-y}}{y} \int_{0}^{+\infty} e^{-x/y} \, dx = e^{-y}$.