Introduction to Probability Models

Lecture 31

Qi Wang, Department of Statistics

Nov 6, 2017

Example 1

The weekly amount spent for maintenance and repairs at a certain company has an approximately normal distribution with a mean of 650 dollars and a standard deviation of 35 dollars.

  1. What is the probability that the company spends less than 675 dollars on maintenance and repairs in one week?
  2. If 725 dollars is budgeted to cover the maintenance/repairs for next week, what is the probability that the actual cost will exceed the budgeted amount?
  3. For planning purposes, the company wants to know the range for the middle 60% of the distribution of weekly maintenance and repair costs. Find the values that determine the middle 60% of the distribution of maintenance/repair costs.
  4. What should the company expect their maintenance/repair costs to be for a year? .

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 2

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 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?