The two samples are independent also let x y s2x s2y be

Question

Let X₁, X₂, X₃, X₄ be iid random variables each one with probability N(10, 4σ).

Let Y₁, Y₂, Y₃ be iid random variables each one with probability N(5, 3σ).

The two samples are independent. Also, let ???X?¯???? , ???Y?¯????, S²_X, S²_Y be the corresponding sample means and sample variances of the two samples.

Answer the following questions:

a. What is the distribution of ???X?¯??-?Y?¯???? ?

b. What is the distribution of ??16σ?2????3S?X?2????+?9σ?2????2S?Y?2????? ?

For (a) I got that ???X?¯??-?Y?¯???? ~ N[5, ??σ√?7????? ].