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 {X_{1}, ...X_{n}} is a sample of n observations of X. Then, the random variable Y defined by
Y = (X_{1} + ... + 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} = [(X_{1}  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 chisquare 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
(n  1) * S^^{2} / ø^^{2} = 9 * 7.6 / 4 = 17.1
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.
