Tutorial

• exponential distribution
  • if $X\sim \exp (\lambda )$, then
    • $f(x)=\lambda e^{-\lambda x}$ $x\ge 0$, $E(X)=1/\lambda$, $Var(X)=1/{\lambda }^2$
    • memoryless property: $P_X(X>s+t|X>s)=P_X(X>t)$
  • 👨‍🏫 $X$: waiting time for first arrival
  • shares the parameter with Poisson distribution
  • as expected time to wait: $1/\lambda$

Exercises

1. continuous r.v.
   • for some constant $c$, pdf of $X$ is: $$f(x)=\{\begin{array}{ll}c{x}^n& 0<x<1\\ 0& \text{otherwise}\end{array}$$
   • find $c$, $P(X>x),0<x<1$
   • when $n=4$, find $E(X)$ and $Var(X)$
   • ${\int }_0^{1}c{x}^{n}dx$
2. normal dist.
   • total sales: follow normal distribution
   • $\mu =2,500,000$, $\sigma =300,000$
   • probability: firm's sale fall within 150,000 of the mean?
     • $2\cdot (\Phi (0.5)-0.5)=2\cdot (0.6915-0.5)=0.383$
   • determine: sales level w/ 9% of being exceeded next year
     • $z_{0.91}=1.34$
     • $2,500,000+1.34\cdot 300,000=2902000$
3. normal dist.
   • weights of kindergarten students: $\sim N(19.5,3.2^2)$
   • find probability: exactly $4/11$ randomly selected students weigh less than $20$
   • $P(X\le 20)=P(Z\le 0.5/3.2)=\Phi (0.156)=0.5636$
   • and $Y\sim Bin(11,0.5623)$
     • $P(Y=4)=\left(\genfrac{}{}{0ex}{}{11}{4}\right)(0.5623{)}^4(1-0.5624{)}^7=0.1014$
4. exponential dist.
   • $X$: exponential r.v. w/ $\mu =1/\lambda$
   • show: $E(X^k)=\frac{k!}{{\lambda }^k}$ $$\begin{array}{rl}E(X^k)& ={\int }_{-\infty }^{\infty }{X}^k\cdot \lambda e^{-\lambda x}dx\end{array}$$
5. Chi-squared dist.
   • if $X\sim N(0,1)$, find pdf of $X^2$
   • $p(x)=2\cdot \frac{1}{\sqrt{2\pi }}{e}^{-0.5x}$
     • pdf of $X\sim N(0,1)$ times two, as two $\sqrt{x}$ exists with same probability
     • ${\chi }^{\prime }=[0,\infty )$