Revision: Differentiation

Lecture 4: Normal distribution

In today's lecture, we learnt two approximations.

1. Binomial distribution to normal distribution

If X~B(n, p)

• Recall that if when n is large, and np is smaller than 5, we can approximate X to Poisson distribution with mean np, i.e. X~Po(np)
• If n is large, np is larger than 5 and n(1-p) is also larger than 5, we can approximate X to Normal distribution with mean np and variance np(1-p), i.e. X~N(np, np(1-p)

2. Poisson distribution to normal distribution

If \[X \sim Po(\lambda)\], and \[\lambda>5\],

we can approximate X to normal distribution, \[X \sim N(\lambda,\lambda)\]

Remark: Remember to do the continuity correction when you approximate either Binomial or Poisson distribution to Normal distribution.

You can attempt tutorial 15B now.

Lecture 3: Normal distribution

In today's lecture, we learnt three important properties of Normal distribution.

1. Sum of two independent normal random variables

2. Sum of n independent observations from the same Normal distribution

3. Multiple of Normal variables.

Remember the following rules:

• All the random variables must be independent.
• When two random variables are added or subtracted, their variance are always added.
• When a random variable is multiplied by a constant, say, a, the variance is multiplied by a^2.
• You can only add or multiply the variance, NOT the standard deviation.

Lecture 2: Normal distribution

In this lecture, you have learnt the following points:

1. the use of " invNorm"

Example: X~ N(17,11), P(X is smaller than a ) =0. 38, use invNorm (0.38, 17, sqrt(11)) to find the value of a.

Note: If the given information is P(X is bigger than a)=0.38, you have change it to P(X is smaller than a)=1-0.38=0.62 before you use the "invNorm" function.

2. Properties of Normal distribution

If \[X\sim N(\mu ,\sigma ^{2}) \], \[aX+b \sim N(a\mu+b, a^{2}\sigma^{2})\]

3. Standard Normal distribution

• Standard Normal distribution is the normal distribution with mean 0 and variance 1.
• We can change any normal distribution to standard normal distribution. \[X\sim N(\mu ,\sigma ^{2}) \], \[\frac{X-\mu}{\sigma}\sim N(0,1)\]

You can make use the properties of Normal distribution to prove it.

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)

Lecture 4:Binomial and Poisson distribution

In this lecture we learnt two important ideas.

1. Find the mode for Poisson distribution

• Similar to Binomial distribution, the Poisson distribution has probabilities that increase to a certain level and decrease subsequently.
• Use G.C. to find the probabilities for each value of X and look for the one with highest probability.
• The mode is usually near the expectation.

2. Additive properties of Poisson Random variable

• If X~Po(a) and Y~Po(b) , where X and Y are independent, then

X+Y ~ Po (a+b)

After this lecture, you can attempt Q1,2,3,4 (except (v)) ,Q5, Q6, Q7.