## Introduction to Probability Models

Lecture 31

Qi Wang, Department of Statistics

Nov 5, 2018

### Announcement

• Quiz 7 has been rescheduled! The new date is Nov, 14.

### Combining Normal Distributions

If we have independent Normal random variables, then the sum(or other linear combination) of these Normal random variables is ALSO Normal

If $X_1 \sim N(\mu_1, \sigma_1), X_2 \sim N(\mu_2, \sigma_2), \cdots, X_n \sim N(\mu_n, \sigma_n)$, and $X = \sum_{i=1}^n X_i$, then

• $X \sim N(\mu, \sigma)$
• $\mu = E[X] = \sum_{i = 1}^n \mu_i$
• $Var(X) = \sum_{i=1}^n \sigma_i^2$
• $\sigma = SD(X) = \sqrt{Var(X)} = \sqrt{\sum_{i=1}^n \sigma_i^2}$

### Example 1

Let $X_1, X_2$ and $X_3$ be independent Normal random variables, where $$X_1 \sim N(\mu = 4, \sigma = 2), X_2 \sim N(\mu = 3.1, \sigma = 7), X_3 \sim N(\mu = 1.5, \sigma = 1.4)$$

1. If $Y = X_1 + X_2 + X_3$, then what is the distribution of $Y$? Find Find the $83_{rd}$ percentile of $Y$
2. Let $K = 2X_3 - X_2 + \frac{1}{3} X_1$, What is the distribution of $K$

### Normal Approximation to the Binomial

If a Binomial distribution has a large enough combination of n and p, it behaves much like a Normal distribution, which means we can use the Normal distribution to approximate the original Binomial distribution

• If $X \sim Bin(n, p)$, and $np > 5, n(1 - p) > 5$
• Then we can use $X^\star \sim N(\mu = np, \sigma = \sqrt{np(1-p)})$, to approximate $X$

You may notice that Binomial is Discrete, and Normal is Continuous. This means the approximation comes at a cost of accuracy that we must try to correct. When we use the approximation, we need to perform a continuity correction:

• If you’re looking for: $P(a \le X \le b)$
• Use $P(a - 0.5 < X^\star < b + 0.5)$

### Example 2

If all conditions are satistified, find the Normal approximation to the following probability statement where $X$ follows a Binomial distribution

1. $P(4 \le X \le 10)$
2. $P(4 < X < 10)$
3. $P(X \le 6)$
4. $P(X < 5)$
5. $P(X \ge 9)$
6. $P(X > 8)$

### Example 3

A class has 400 students, and each drops the course independently with probability 0.07. Let X be the number of students that finish the course

1. Find $P(370 \le X \le 373)$, what is the exact distribution of $X$?
2. Any approximation?

### Three Approximations in this Course

1. The Binomial approximation to the Hypergeometric: If $X \sim HyperGeom(N, m, n)$, and $N > 20n$, we can use $X^\star \sim Binomial(n = n, p = \frac{m}{N})$, to approximate $X$
2. The Poisson approximation to the Binomial: If $X \sim Bin(n, p)$ with $n>100$ and $p<0.01$, we can use $X^\star \sim Poisson(\lambda = np)$, to approximate $X$
3. The Normal approxmation to the Binomial: If $X \sim Bin(n, p)$, and $np > 5, n(1 - p) > 5$, then we can use $X^\star \sim N(\mu = np, \sigma = \sqrt{np(1-p)})$, to approximate $X$