Question

For n = 1, 2, 3, …, let the joint moment generating function of (X, Y_n) be
\(M_{X,Y_n}(t_1,t_2)=e^{\frac{t^2_1}{2}(1-2t_2)^{-\frac{n}{2}}}, t_1 \in \R,t_2 \lt \frac{1}{2}.\)
If \(T_n=\frac{\sqrt{n}X}{\sqrt{Y_n}},n \ge1,\) then which one of the following statements is true ?

• A. The minimum value of n for which Var(T_n) is finite is 2
• B. \(E(T^3_{10})=10\)
• C. \(Var(X+Y_4=7)\)
• D. \(\lim\limits_{n \rightarrow \infin}P(|T_n|>3)=1-\frac{\sqrt2}{\sqrt{\pi}}\int^3_0e^{-\frac{t^2}{2}}dt\)

Answer 1

The Correct Option is D

The correct option is (D) : \(\lim\limits_{n \rightarrow \infin}P(|T_n|>3)=1-\frac{\sqrt2}{\sqrt{\pi}}\int^3_0e^{-\frac{t^2}{2}}dt\).