Question: Show that if X1 X2 Xn
Show that if X1, X2, . . . , Xn are independent random variables having the chi-square distribution with v = 1 and Yn = X1 + X2 + · · · + Xn, then the limiting distribution of
As n → ∞ is the standard normal distribution.
Answer to relevant QuestionsBased on the result of Exercise 8.24, show that if X is a random variable having a chi– square distribution with v degrees of freedom and v is large, the distribution of X – v / √2v can be approximated with the ...With reference to Exercise 8.2, show that if the two samples come from normal populations, then 1 – 2 is a random variable having a normal distribution with the mean µ1 – µ2 and the variance σ21/n1 + σ22/n2. If X has an F distribution with v1 and v2 degrees of freedom, show that Y = 1/X has an F distribution with v2 and v1 degrees of freedom. Find the sampling distributions of Y1 and Yn for random samples of size n from a population having the beta distribution with α = 3 and β = 2. Use the result of Exercise 8.56 to find the mean and the variance of the sampling distribution of R for random samples of size n from the continuous uniform population of Exercise 8.46.
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