Lecture 31

Qi Wang, Department of Statistics

Nov 6, 2017

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.

- What is the probability that the company spends less than 675 dollars on maintenance and repairs in one week?
- 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?
- 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.
- What should the company expect their maintenance/repair costs to be for a year? .

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

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

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

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

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 youâ€™re looking for: $P(a \le X \le b)$
- Use $P(a - 0.5 < X^\star < b + 0.5)$

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

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