Suppose X1;X2;...;X5 is a random sample from a n(0; variance) distribution. define
U = X1+X2+X3+X4;
V = (X2)^2+(X3)^2+(X4)^2+(X5)^2
and W = U/sqrtV.
(a) Name the distribution of U/4 and V/(variance) and give the values of their parameters.
(b) (i) Compute k such that kW has a t-distribution.
(ii) Give the number of degrees of freedom of this t-distribution.
(c) Compute E(V).
To answer these questions, we need to understand the properties of the given random variables U, V, and W, and also derive their distributions.
(a) The distribution of U can be obtained by summing the random variables X1, X2, X3, and X4. Since each X is a random sample from a normal distribution with mean 0 and variance variance, the sum U follows a normal distribution. The mean of U is 0 + 0 + 0 + 0 = 0, and the variance of U is variance + variance + variance + variance = 4 * variance. Therefore, U/4 follows a normal distribution with mean 0 and variance variance.
The distribution of V can be obtained by summing the squared random variables X2, X3, X4, and X5. Since each Xi^2 is a random variable following a chi-squared distribution with 1 degree of freedom (since Xi is from a normal distribution), the sum V follows a chi-squared distribution with 4 degrees of freedom. The parameter for this chi-squared distribution is the number of degrees of freedom, which in this case is 4.
Therefore, U/4 follows a normal distribution with mean 0 and variance variance, and V/(variance) follows a chi-squared distribution with 4 degrees of freedom.
(b) (i) To find k such that kW follows a t-distribution, we need to determine the degrees of freedom for this t-distribution. Since V follows a chi-squared distribution with 4 degrees of freedom, and U/4 follows a normal distribution, W is the ratio of two independent random variables. By the properties of the t-distribution, this ratio follows a t-distribution. Therefore, k is given by the square root of the ratio of the degrees of freedom of V and the variance of U/4.
k = sqrt(4 / variance)
(ii) The number of degrees of freedom for this t-distribution is equal to the degrees of freedom of the chi-squared distribution (i.e., the denominator of W).
The number of degrees of freedom for the t-distribution is 4.
(c) To calculate E(V), we need to find the expected value of the chi-squared distribution with 4 degrees of freedom. The expected value of a chi-squared distribution with k degrees of freedom is k.
E(V) = 4
Therefore, E(V) = 4.