Let X 1, ......, X m N( 1 , )and Y 1 , . . .,Y n N( 2 , ) be independent

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

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