Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Random Variables & Distribution Functions
Let {Xn : n ≥ 1} be a sequence of independent and identically distributed random variables and the probability mass function of X1 is the following;
P(X1 = 1) = P(X1 = 3) = \(\frac{1}{2}\).
If Yn = X1 + ··· + Xn, then which of the following staements are correct ?
1
\(\frac{Y_n}{n}\) converges to 2 in probability.
2
Variance \(\left(\frac{Y_n}{n^{2 / 3}}\right)\) converges to 0, as n → ∞.
3
\(\frac{Y_n}{n^{2 / 3}}\) converges to c in probability, where 0 < c < 0.
4
\(\frac{Y_n}{n^2}\) converges to 0 in probability.