Chapter 5.6 From DeGroot & Schervish

Uniform Distribution

Mean and Variance

The Normal Distributions The most widely used model for random variables with continuous distributions is the family of normal distributions. the random variables studied in various physical experiments often have distributions that are approximately normal. If a large random sample is taken from some distribution, many important functions of the observations in the sample will have distributions which are approximately normal.

Properties of Normal Distributions A random variable X has the normal distribution with mean μ and variance σ 2 (−∞ 0) if X has a continuous distribution with the following p.d.f.:

The m.g.f. of Normal Distribution

Mean and Variance The mean and variance of the normal distribution are μ and σ 2, respectively. The first two derivatives of the m.g.f. of normal distribution

The Shapes of Normal Distributions The p.d.f. f (x|μ, σ 2 ) of the normal distribution with mean μ and variance σ 2 is symmetric with respect to the point x = μ. Therefore, μ is both the mean and the median of the distribution. The p.d.f. f (x|μ, σ 2 ) attains its maximum value at the point x = μ.

The Shapes of Normal Distributions

Linear Transformations If a random variable X has a normal distribution, then every linear function of X will also have a normal distribution. Theorem If X has the normal distribution with mean μ and variance σ 2 and if Y = aX + b, where a and b are given constants and a ≠0, then Y has the normal distribution with mean aμ + b and variance a 2 σ 2.

Linear Transformations Proof If ψ denotes the m.g.f. of X, if ψ Y denotes the m.g.f. of Y, then By comparing this expression for ψ Y with the m.g.f. of a normal distribution given, we see that ψ Y is the m.g.f. of the normal distribution with mean aμ + b and variance a 2 σ 2. Hence, Y must have this normal distribution.

The Standard Normal Distribution

Theorem

Converting Normal Distributions to Standard Let X have the normal distribution with mean μ and variance σ 2. Let F be the c.d.f. of X. Then Z = (X − μ)/σ has the standard normal distribution, and, for all x and all 0<p <1, Proof Z = (X − μ)/σ has the standard normal distribution. Therefore, For second equation, let p = F(x) in the first equation and then solve for x in the resulting equation.

Example

Comparisons of Normal Distributions

Linear Combinations of Normally Distributed Variables Theorem If the random variables X1,..., Xk are independent and if Xi has the normal distribution with mean μi and variance σ 2 i (i = 1,..., k), then the sum X Xk has the normal distribution with mean μ μk and variance σ σ 2 k.

Linear Combinations of Normally Distributed Variables

Corollary

Example Suppose that the heights, in inches, of the women in a certain population follow the normal distribution with mean 65 and standard deviation 1, and that the heights of the men follow the normal distribution with mean 68 and standard deviation 3. Suppose also that one woman is selected at random and, independently, one man is selected at random. Determine the probability that the woman will be taller than the man.

Example Let W denote the height of the selected woman, and let M denote the height of the selected man. Then the difference W −M has the normal distribution with mean 65 − 68=−3 and variance = 10. Therefore, if we let then Z has the standard normal distribution. It follows that

Corollary

Example Suppose that a random sample of size n is to be taken from the normal distribution with mean μ and variance 9. Determine the minimum value of n for which

Example the sample mean Xn will have the normal distribution for which the mean is μ and the standard deviation is 3/n 1/2. Therefore, if we let then Z will have the standard normal distribution. In this example, n must be chosen so that

Example

The Lognormal Distributions If log(X) has the normal distribution with mean μ and variance σ 2, we say that X has the lognormal distribution with parameters μ and σ 2.

m.g.f. of a Lognormal Distribution The moments of a lognormal random variable are easy to compute based on the m.g.f. of a normal distribution. The definition of ψ is ψ(t) = E(e tY ). Since Y = log(X), we have It follows that E(X t ) = ψ(t) for all real t. In particular, the mean and variance of X are