Sampling Distributions: Chi-Square, Student's t & F-Distributions
“Degrees of freedom, ratios of Chi-squares, heavy tails, and small-sample statistics.”
1.The Geometry of Sampling Distributions
§ 4.1.Statistical inference relies on probability distributions derived from functions of sample statistics.
1. Chi-Square Distribution ChiSq(k):
Arises when summing the squares of k independent standard normal random variables. It is right-skewed and defined strictly on [0, infinity).
Mean = k, Variance = 2k.
2. Student's t-Distribution t(k):
Formulated by William Gosset ('Student') in 1908 to account for additional uncertainty when sample size is small and the true population variance sigma^2 is unknown and replaced by sample variance s^2.
T = [ (X_bar - mu) / (s / sqrt(n)) ] ~ t(n - 1).
3. Snedecor's F-Distribution F(d_1, d_2):
Used to compare variances between two independent populations (ANOVA tests).
2. Worked Practice Problem: Identifying Sampling Distribution Constructs
Ensure the denominator chi-square variable is divided by its own degrees of freedom before taking the square root.
Problem: Let Z1, Z2, Z3, Z4, Z5 be independent standard normal random variables ~ N(0, 1). Identify the exact distribution (including parameter values) of: W = Z1 / sqrt[ (Z2^2 + Z3^2 + Z4^2 + Z5^2) / 4 ].
Step 1: Inspect the numerator:
Z1 ~ N(0, 1) is a single standard normal variable.
Step 2: Inspect the sum in the denominator:
V = Z2^2 + Z3^2 + Z4^2 + Z5^2
Since Z2, Z3, Z4, Z5 are 4 independent standard normals, the sum of their squares follows a Chi-square distribution with 4 degrees of freedom: V ~ ChiSq(4).
Step 3: Check independence between numerator and denominator:
Z1 is independent of Z2, Z3, Z4, Z5, so Z1 and V are independent.
Step 4: Apply Student's t-distribution definition:
T = Z / sqrt(V / k), where Z ~ N(0, 1), V ~ ChiSq(k), and Z, V are independent.
Here k = 4.
Conclusion: W follows a Student's t-distribution with 4 degrees of freedom: W ~ t(4).
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