Introduction to Probability Models

Lecture 31

Qi Wang, Department of Statistics

Nov 5, 2018


  • 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 $X$