Let X1X2X3 be a sequence of independent random variables suc

Let X1,X2,X3,... be a sequence of independent random variables such that Xk is Binomial with parameters k and p. define

show that the sequence Mn converges in probability and find the limit.

Solution

Mn= (x1+x2+ .......+xn) / ( n(n+1)/2 )

(X1 + X2 + · · · Xn) has a Binomial(1 + 2 + · · · + n, p) distribution.
E( X1 + X2 + · · · Xn)= (1 + 2 + · · · + n)* p = n(n+1)p / 2

so, E(Mn) = p VAR(x1+...+xn)=n(n+1)pq/2
so, var (Mn) = pq / (n(n+1) /2 ) which tends to 0 as n tends to infinity.

p[ | Mn - E(Mn) | >= ] <= v(Mn)/^2 [using chebyshev-markov\'s WLLN]
= pq/(n(n+1)/2)^2 which tends to as n tends to infinity.

thus, Mn converges in prob. to p.