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 $X$