## Introduction to Probability Models

Lecture 32

Qi Wang, Department of Statistics

Nov 8, 2017

### Example 1

Heights of Pokémon are Normally Distributed with a mean of 59 inches and a standard deviation of 17 inches.

1. What is the standardized height (z-score) for Blastoise, who is 63 inches tall?
2. Using the Normal Table, find the probability that any Pokémon is taller than Blastoise
3. Knowing that a Pokémon is taller than Blastoise, what is the probability that it is taller than 70 inches?
4. What height corresponds to the top 10% of Pokémon heights?

# Time for Quiz

### 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. $P(X > 8)$

### 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 $X$