Let X 1, ......, X m N( 1 , )and Y 1 , . . .,Y

Let X_1,......, X_m ∼ N(μ₁, σ)and Y₁, . . .,Y_n  ∼ N(μ₂, σ) be independent random samples from the specified normal population distributions (note that the population sd’s are equal). Let S²₁ and S²₂ denote the sample variance of the two samples, and define a pooled variance estimator of σ² by

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m-1 m+n-2 n-1 -S₁+ -S²/ m+n-2 Show that (m+n-2)/² has a chi- squared distribution with m + n - 2 df.