機率數學問題-請各位大哥幫幫我？？

1. An urn contains five balls,two of wich are maked $1,two $5,and one $15. A game is played by paying $10 for winning the sum of the amounts maked on two balls selected randomly from the urn. Is this a fair game?

[R]
令 X1, X2 為先後抽出的球上面的金額, 則
E[X1+X2] = E[X1]+E[X2] = 2E[X1] = 2{1*2/5+5*2/5+15*1/5}=54/5
付 $10 玩一次期望值 $54/5 > $10. 對玩家有利, 非公平賭局.


2. The distribution function of a random variable X is given by

F(x)={
0 if x<-3
3/8 if -3<=x<0
1/2 if 0<=x<3
3/4 if 3<=x<4
1if x>=4
Calculate E(X),E(X^2-2|X|),and E(X|X|)

[R]
What's "E[X|X|]"? 是 X(|X|) 的期望值?
P[X=-3] = 3/8,
P[X=0] = 1/2 -3/8 = 1/8
P[X=3] = 3/4-1/2 = 1/4
P[X=4] = 1-3/4 = 1/4
期望值請自己算!


3.Let X be a random integer from the set {1,2,....,N}. Find E(X), Var(X), and δx

[R]
"δx" 是啥?
E[X] = 1(1/N)+2(1/N)+...+N(1/N) = (N+1)/2
E[X^2] = 1^2(1/N)+...+N^2(1/N) = [N(N+1)(2N+1)/6](1/N)
　　　= (N+1)(2N+1)/6
Var[X] = E[X^2]-(E[X])^2, 自己算.



4.Suppose that Xis a discrete random variable with E(X) = 1 and E[X(X-2)]=3 Find Var( -3X+5).

[R]
Var[-3X+5] = (-3)^2 Var(X) = 9Var(X)
E[X^2] = E[X(X-2)]+2E[X],
其餘計算自己來.



5.Let X be the amount (in fluid ounces ) of soft drink ...

[R]
然後? 問啥?


6.A random variable X has the density function

f(x)={
3e^(-3x) if 0<=x<無限大
0otherwire
Calculate E(e^X).

[R]
E[e^X] = ∫_[0,∞) e^x 3e^{-3x} dx = 3/2


7.Let the probability density function of tomorrow’s Celsius temperature be h.. In terms of h , calculate the corresponding probability density function and its expection for Fahrenheit temperature.
Hint: Let C and F be tomorrow’s temperature in Celsius and Fahrenheit, respectively. Then F = 1.8C +32

[R]
Let X be the temperature in Celsius scale, and
let Y be the temperature in Fahrenhei scale.
Then Y = 1.8X+32, so X = (Y-32)/1.8.
P[Y =< y] = P[X =< (y-32)/1.8] = ∫h(x) dx
The p.d.f. for Y is
　 g(y) = (1/1.8)h((y-32)/1.8)

(看不懂? 對照課本看吧!)


8./9.
題目不全, 如 5.

如果都是中文題就好做多了~