Week 10. Continuous Random Variable-II

Normal Random Variable

• X is a normal random variable with E(X)= $\mu \in \mathcal{R}$ and Var(X)= $\sigma^2 >0$
  $$ X \sim N(\mu, \sigma^2)$$
• pdf f(x), cdf F(X)
• $$ X \sim N(\mu, \sigma^2) \Rightarrow \frac{X-\mu}{\sigma} \sim N(0,1).$$ A N(0,1) random variable, i.e. a normal random variable with mean zero and variance 1, is commonly called a standard normal and denoted by Z
• pdf and cdf of Z ~ N(0,1). Their definitions and properties
  • pdf
    $$ \phi(x) = \frac{1}{\sqrt{2 \pi }} e^{- x^2/2} $$
  • cdf
    $$ \Phi(x) = \int_{-\infty}^x \phi(t) dt.$$
  • Properties

Homework 5

Textbook Sec 5.8: Quick Exercises: 5.6, 5.7; Exercises: 5.13, 5.14. To be discussed in class 11/22.