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) = P (a ≤ X ≤ b) = F (b) − F (a).

Definitions 0.2. Z is a Standard Normal random variable, (denoted Z ∼ 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.

Now, the book uses a different function

I(t) = \frac{1}{\sqrt{2π}} ∫_{−∞}^{t} e^{−x^{2}/2}dx = Φ(t) − \frac{1}{2}

for t ≥ 0.

In other words:

Φ(t) = \frac{1}{2} + I(t) for t ≥ 0

Φ(t) = \frac{1}{2} − I(t) for t < 0

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 → ∞.

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 → ∞.

Problems

1. Find P (−1 < X < 2), P (|Z| < 1), P (|Z| < 2) and P (|Z| < 3), P (Z > 1.5).

2. Find t such that (a) P (Z > t) = .95,

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

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

(a) Find the mean and variance of Y.

(b) Find approximate values for the mass distribution of Y.

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.

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?

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.

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?

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.