Solving T-Distribution Problems: Finding the Value of 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.

(to moderator: I totally forgot about not posting hmk problems in this section.)

 

[Valkarie]

In order to solve this problem, you need to make use of the fact that a t-distribution can be expressed as a scaled and shifted version of a normal distribution. That is, if X has a normal distribution with mean μ and variance σ2, then Z = (X-μ)/σ has a t-distribution with n degrees of freedom. From this, we can see that in order for T to have a t-distribution, we must set K = 1/σ, where σ is the standard deviation of the normal distributions of X, Y, Z and W. Since all of these variables have mean 0 and variance > 0, their standard deviations are equal to their variances. Therefore, we can calculate σ as the square root of the variance of any one of these variables, say W, and set K = 1/σ. Since there are 4 independent normal variables in T, it follows that T has a t-distribution with n = 4 - 1 = 3 degrees of freedom.

 

[tacman]

To solve this problem, we first need to understand the properties of a Student's t-distribution. A Student's t-distribution is a probability distribution that is used to estimate the mean of a normally distributed population when the sample size is small (less than 30) or when the population standard deviation is unknown. It is similar to a normal distribution, but has heavier tails and a flatter peak.

To find the value of K, we need to use the following formula:

T=K(X+Y)/(Z^2 + W^2)^1/2

Since X, Y, Z, and W are independent normal variables with mean 0 and variance >0, we can rewrite the formula as:

T=K(X+Y)/√(Z^2 + W^2)

We know that a Student's t-distribution has a mean of 0 and a variance of n/(n-2), where n is the degrees of freedom. So, we can set the mean and variance of T equal to these values:

Mean of T = 0 = K(0+0)/√(0+0) = 0

Variance of T = n/(n-2) = K^2(Var(X)+Var(Y))/Var(Z+W) = K^2(2Var(X))/2Var(Z) = K^2

Since the variance of T is equal to K^2, we can solve for K:

K^2 = n/(n-2)

K = √(n/(n-2))

Therefore, the value of K is √(n/(n-2)).

To find the degrees of freedom, we need to solve for n:

K^2 = n/(n-2)

nK^2 - 2K^2 = n

n(K^2 - 2) = 2K^2

n = 2K^2/(K^2 - 2)

Substituting our value of K, we get:

n = 2n/(n-2-2)

n = 2n/n

n = 2

Therefore, the degrees of freedom for T is 2.

In summary, to find the value of K and degrees of freedom for a Student's t-distribution, we need to set the mean and variance of the distribution equal to the mean and