【学习笔记】数理统计习题一

Q1: The planet Tralfamadore has years with 500 days. There are 5 Tralfamadorans in the room. Write an expression for the probability that no two of them have the same birthday.

Solution: Suppose that there are n Tralfamadorans in the room. The probability that no two of them have the same birthday is given by
$$p(n):=\{\begin{array}{ll}\frac{A_{500}^n}{500^n},& n\le 500\\ 0,& n>500\end{array}$$
When $n=5$， we have $p(5)=A_{500}^5/500^5\approx 0.9801$

How would you find the smallest $n$ for which a room of $n$ Tralfamadorans has probability at least $1/2$ of having two members with the same birthday?

Solution: $n$要满足的条件为
$$1-\frac{500!}{(500-n)!500^n}\ge \frac{1}{2}$$
即
$$\frac{500!}{(500-n)!500^n}\le \frac{1}{2}$$
可计算得$n$的最小值为27

The above two questions really require some sort of assumption to get an answer. In case you did not already provide one, what is the customary assumption one uses in probability exercises?

Solution: The birthdays of the $n$ Tralfamadorans are independent and uniformly distributed over the 500 days.

Q2: Write an expression for ${\varphi }_X(t)$, the moment generating function (MGF) of a random variable X. Find and interpret the second derivative $\varphi {{}^{\prime \prime }}_X(0)$. If the MGF does not exist what would we use instead?

Solution:
$${\varphi }_x(t)={\int }_{-\infty }^{+\infty }{e}^{tx}dF(x)$$

$${\varphi }_x^{{}^{\prime \prime }}(0)={\int }_{-\infty }^{+\infty }{x}^2{e}^{tx}dF(x){|}_{t=0}=E{x}^2$$

​ 用特征函数表示
$${\varphi }_x^{{}^{\prime \prime }}(0)=E{x}^2=-f^2(0)$$

Q3: If $X$ and $Y$ are uncorrelated random variables must they be independent?If $X$ and $Y$ are independent random variables must they be uncorrelated?Explain in both cases.

Solution: 如果$X$和$Y$是不相关的随机变量，它们不一定是独立的，比如说，存在一个简单的例子，

X \ Y	-1	0	1
0	0	1/3	0
1	1/3	0	1/3

从表格中可以得知，$E\xi =0$, $E\eta =E{\xi }^2=\frac{2}{3}$, $E\xi \eta =E{\xi }^3=0$，故$cov(\xi ,\eta )=E\xi \eta -E\xi \cdot E\eta =0$，所以$\xi$与η\eta