练习

Suppose and are random variables on and let . Show that if we let for and for , then is a random variable.

证明

对任意 Borel 集 ，

从而 也是随机变量。

练习

Let have the standard normal distribution. Use Theorem 1.2.6 to get upper and lower bounds on .

证明

练习

Show that a distribution function has at most countably many discontinuities.

证明

由于分布函数是单调有界函数，从数学分析相关知识知道其不连续点一定都是跳跃间断点，且不连续点至多可数。

练习

Show that if is continuous then has a uniform distribution on , that is, if .

证明

对 ，

练习

Suppose has continuous density , and is a function that is strictly increasing and differentiable on . Then has density

for and otherwise. When with ,

so the answer is

证明

对 ，有

令 换元得到

练习

Suppose has a normal distribution. Use the previous exercise to compute the density of . (The answer is scrled the lognormal distribution.)

证明

令 得到 的概率密度函数

练习

  1. Suppose has density function . Compute the distribution function of and then differentiate to find its density function.

  2. Work out the answer when has a standard normal distribution to find the density of the chi-square distribution.

证明

  1. 记 的分布函数为 ，那么

    即为其分布函数。记其概率密度函数为 ，则

  2. 代入正态分布概率密度函数得到