Solving T-Distribution Problems: Finding K and Degrees of Freedom

[matrix_204]

I had a homework problem which I'm having trouble with. Unfortunately I missed my class when the teacher was giving examples of t-distribution problems. If someone could help me about the steps involving to solve this problem, I would really appreciate it.

Let T=K(X+Y)/(Z^2 + W^2)^1/2 where X,Y,Z and W are independent normal variables with mean 0 and variance >0. Find the value of K so that T has a student's t distribution. How many degrees of freedom does T have?

One of my friends had told me that K=1 and (d.o.f.) n=2 for T, but unable to give a proper explanation of how he got it. I just want to know the steps to solve this.

Thank you.

 

[lippyka]

To solve this problem, the first step is to calculate the variance of T. Since X, Y, Z, and W are independent normal variables, their variances can be added together. Thus, the variance of T is K^2 * (Var(X) + Var(Y)) / (Z^2 + W^2)^2. Next, we need to make sure that T has a Student's t distribution. To do this, we need to make sure that the variance of T is equal to the variance of a Student's t distribution with some degrees of freedom. We can find this by using the equation Var(T) = n/(n-2), where n is the number of degrees of freedom. Therefore, we can set the two equations equal to each other, and solve for K and n: K^2 * (Var(X) + Var(Y)) / (Z^2 + W^2)^2 = n/(n-2) After simplifying this equation, we find that K = sqrt((n-2) * (Z^2 + W^2)^2) / sqrt(Var(X) + Var(Y)) and n = 2. Therefore, the value of K is sqrt((2-2) * (Z^2 + W^2)^2) / sqrt(Var(X) + Var(Y)) = 0 and the number of degrees of freedom is 2.