MATH 367, spg 98, CHAPTER 6 Section 3, Apr 1: Normal Distribution

Definitions 0.1. X is a continuous random variable with density function f(x) if 1. f(x) ≥ 0

2. ∫_{−∞}^{∞} f(x)dx = 1

3. P (a < X < b) = P (a ≤ X ≤ b) = ∫_{a}^{b} f(x)dx.

In such case we have E(X) = ∫_{−∞}^{∞} xf(x)dx and E(h(X)) = ∫_{−∞}^{∞} h(x)f(x)dx. The function F (x) = P (X < x) = ∫_{−∞}^{x} f(x)dx is called the Distribution Function of X. One then has P (a < X < b) = F (b) − F (a).

Definitions 0.2. Z is a Standard Normal random variable, (denoted X ∼ N(0, 1)), if its density function is

φ(t) = \frac{1}{\sqrt{2π}} e^{−t^{2}/2}.

The standard normal has mean 0 and standard deviation 1. This density function of the standard normal does not have a close formula for integrating it. Thus one needs to use tables that estimate its distribution function Φ(t) = ∫_{−∞}^{t} e^{−x^{2}/2}/\sqrt{2π}dx. Thus one can always estimate P (a < Z < b) = Φ(b) − Φ(a) using the tables. Properties:

1. φ is symmetric: φ(−x) = φ(x).

2. Φ(−x) = 1 − Φ(x).

3. Φ(0) = 1/2.

4. E(Z) = 0 and Var(Z) = E(Z^{2}) = 1.

Problem 1: Find P (−1 < X < 2), P (|Z| < 1), P (|Z| < 2) and P (|Z| < 3), P (Z > 1.5). Problem 2: Find t such that

1. P (Z > t) = .95,

2. P (|Z| > t) = .95.

Theorem 0.3. Central Limit Theorem for the Binomial: DeMoivre–Laplace Theorem. Let X be Binomial B(n, p). Then

P (a < \frac{X − np}{\sqrt{npq}} < b) → Φ(b) − Φ(a)

as n → ∞.

Problem 3: Let X be Binomial B(n, p). Let Y = (X − np)/\sqrt{npq}.

1. Find the mean and variance of Y.

2. Find approximate values for the mass distribution of Y. Problem 4: Let X be the number of times a fair coin, flipped 100 times, lands Heads. Use the normal approximation to compute the probability that X = 50. Problem 5: A die is continually rolled until the total sum of all rolls exceeds 300. What is the probability that at least 80 rolls are necessary?

The Central Limit Theorem holds in many other situations.

Theorem 0.4. Central Limit Theorem: Let X_{1}, X_{2}, … be independent identically distributed (i.i.d.) random variables with mean m and variance σ^{2}. Then

P (a < \frac{X_{1} + … + X_{n} − mn}{σ\sqrt{n}} < b) → Φ(b) − Φ(a)

as n → ∞.

Problem 6: Let X_{1}, X_{2}, … be i.i.d. random variables with mean m and the and variance σ^{2}. Find the mean and the variance of Y = (X_{1} + X_{2} + … + X_{n} − nm)/\sqrt{n}σ. Problem 7: One has 100 lightbulbs whose lifetimes are independent random variables with a mean of 5 hours and variance also 5. If the lightbulbs are used one at a time, with a failed bulb being immediately replaced by a new one, what is the probability that there is still a working bulb after 525 hours? Problem 8: Same as above, but now you have n lightbulbs. Use the Central Limit Theorem to estimate how large need n be so that if S_{n} = X_{1} + … + X_{n}, where X_{i} = lifetime of the ith bulb, P (|S_{n}/n − 5| > .01) < .01

Homework Section 6.3: read section 6.3 and do problems 22–31.