f（x）=
8
<

cx 3，如果x 1

（a）（4分）求常数c。
（b）（3分）计算E（X）。

[0; 1]。令Y为随机变量
Y =
</ s> </ s> </ s> </ s> </ s> </ s>

（c）（4分）计算E（Y）。
（d）（4分）计算Cov（X; Y）。

（a）（4分）使用中心极限定理来估计
S100大于25。
（b）（4分）计算任何n 1.的Cov（Sn; Sn + 1）。
（c）（4分）假设n> 100且事件S100的条件为24。

（d）（4分）使用中心极限定理来估计P（S200 34jS100 =
24）。

（e）（4分）使用中心极限定理估计P（S81> 30）。

Problem 1. Let X be a random variable with probability density function
f(x) =
8
<
:
cx 1=2 if 0 < x  1
cx 3 if x  1
0 otherwise
(a) (4 points) Find the constant c.
(b) (3 points) Calculate E(X).
Now let Z be a random variable, independent of X, with uniform distribution in
[0; 1]. Let Y be the random variable
Y =

Z if X  1
0 otherwise
(c) (4 points) Calculate E(Y ).
(d) (4 points) Calculate Cov(X; Y ).

Problem 2. (15 points) In a game of chance you ip a fair coin with a white side and a black
side a random number of times N. For each coin ip, you win 1 dirham if the white
side turns up and you lose 1 dirham if the black side turns up. Suppose that N is a
Poisson random variable with parameter  > 0. Let X denote your gain or loss in
dirhams at the end of the game and calculate the variance of X.

Problem 3. In a game of chance you ip a fair coin with a white side and a black side. For each
coin ip, you win 1 dirham if the white side turns up and you lose 1 dirham if the
black side turns up. Let Sn denote your gain in dirhams after n coin ips.
(a) (4 points) Use the Central Limit Theorem to estimate the probability that
S100 is larger than 25.
(b) (4 points) Calculate Cov(Sn; Sn+1) for any n  1.
(c) (4 points) Assume that n > 100 and condition on the event S100 = 24.
Calculate E(SnjS100 = 24) and Var(SnjS100 = 24).
(d) (4 points) Use the Central Limit Theorem to estimate P(S200  34jS100 =
24).
Now suppose that you play the same game with a di erent coin. This coin too has
a white side and a black side, but when you ip it, the white side turns up 2 times
out of 3.
(e) (4 points) Use the Central Limit Theorem to estimate P(S81 > 30).

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