Binomial and Poisson distribution Lecture 5+ Normal distribution Lecture 1

1. Binomial and Poisson distribution

For binomial and poisson distribution, we learnt about the approximation binomial distribution to poisson distribution today.

If X~ B(n, p), with n is larger than 50, p is smaller than 0.1 with np smaller than 5, X~ Po(np) approximately.

2. Normal distribution

In today's lecture, we learnt a few things on Normal distribution.

• Continuous random variable

Normal distribution is for continuous random variables. ( Unlike Binomial and Poisson distributions, which are for discrete random variables. )

For a continuous random variable, there is a probability density function (pdf) f(x) associated with it.

However, f(a) does not denote the probability at a. Instead, for continuous random variable, the probability at any specific value is 0. We can only compute the probability of a certain interval, e.g. (a, b). The area under the curve of f(x) from a to b will give the probability P(x is between a and b )

• Important result for Expectation and variance

Expectation

E(a)=a

E(aX+b) = aE(X) + b

E(aX+bY)=aE(X)+bE(Y)

Variance

Var(a) = 0

Var(aX+b)=a^2 Var (X)

Var(aX+bY)= a^2 Var(X)+b^2 Var(Y) if X and Y are independent

• Normal distribution \[X\sim N(\mu, \sigma ^{2})\]

Normal distribution is symmetrical about the mean. The mean, median and mode are all equal.

When you use G.C. to evaluate normal distribution, you should key in:

normalcdf ( lower bound, upper bound, mean, standard deviation)