Suppose you have eight runners run a race; the distribution of their individual run times is Normal and each has mean $11$ seconds, say. The standard deviation of runner one is the smallest, two the second smallest, the the third smallest, etc, and eight the largest. Two questions are confusing m...

Question Let $X_1 , X_2, \cdots , X_n$ be independent and identically distributed with $P(X_i =1)=P(X_i=-1)=p$ and $P(X_i=0)=1-2p$ for all $i=1,2,\cdots,n$. Define $$ a_n=P(\prod_{i=1}^{n}X_i=1), b_n=P(\prod_{i=1}^{n}X_i=-1), c_n=P(\prod_{i=1}^{n}X_i=0)$$ Find $\lim_{n\to\infty} a_n,\lim_{n\to...

Q) Let $X_{1},X_{2},X_{3},....,X_{n}$ be independent and identically distributed with $P(X_{i} = 1) = P(X_{i} = -1) = p$ and $P(X_{i} = 0) =1-2p$ for all $i=1,2,3,...n$. Define $a_{n} = P\left ( \prod_{i=1}^{n} X_{i}=1\right )$$,\;$$b_{n} = P\left ( \prod_{i=1}^{n} X_{i}=-1\right ),\;$$c_{n} =P\...