Probstat/week8 practice 1

This is part of probstat

This exercise considers continuous random variables.

Probability

Uniform distribution

1. Let random variable $X$ is a continuous random variable which is a real number chosen uniformly from the range (10,50). Let $f(x)$ denote its probability distribution function.

• 1.1 Describe $f(x)$.
• 1.2 What is $P\{X>20\}$.
• 1.3 What is $P\{X>20|X<40\}$. (Note that this is a conditional probability.)

2. Let random variable $X$ is a continuous random variable whose probability distribution function $f(x)$ is given by

$f(x)=\left\{{\begin{array}{ll}cx\ \ \ \ &{\mbox{if}}\ 0\leq x\leq 20,\\0&{\mbox{otherwise.}}\end{array}}\right.$

Expectation

Definition: If $X$ is a continuous random variable whose probability distribution function (pdf) is $f(x)$, we define the expected value of $X$ to be

$\mathrm {E} [X]=\int _{-\infty }^{\infty }xf(x)dx$