Egwald Statistics: Probability and Stochastic Processes

by

Elmer G. Wiens

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Distributions | Sampling Distributions | Sample Mean | Hypotheses Testing | Random Walk | References

Normal Distribution Samples

A. Sample Mean

Suppose a random variable X has the normal distribution N(µ, ø^²) and {X₁, ...X_n} is a sample of n observations of X. Then, the random variable Y defined by

Y = (X₁ + ... + X_n)/n

is called the sample mean.

The r.v. Y has normal distribution N(µ, ø^²/n).

Density of the Sample Mean

With µ = 0 ,ø^² = 4, and n = 10

B. Sample Variance

The random variable S^² defined by

S^² = [(X₁ - Y)^² + ... + (X_n - Y)^²] / (n - 1)

is called the sample variance. S is called the standard error of the mean.

The r.v. (n - 1) * S^² / ø^² has the chi-square distribution with n - 1 degrees of freedom.

Suppose that from a sample of ten observations of the population r.v. X, we find that S^² = 7.6. Suppose, further, we want to test if the population normal r.v. X has variance ø^² = 4.

Then

From the graph below, 17.1 > 16.919, so we can reject, with considerable confidence, the hypotheses that the population r.v. X has a variance of 4.