let x1 x2 be a sequence of independent bernoulli random var

let x1, x2, ... be a sequence of independent bernoulli random variables with probabilities p(Xn = 1) = 1/n.

Show that Xn goes 0 as n goes to infinity.

Solution

let x1, x2, ... be a sequence of independent bernoulli random variables with probabilities p(Xn = 1) = 1/n.

Show that Xn goes 0 as n goes to infinity.

ANS- take limit n goes to infinity, p(Xn = 1) = 1/n.= 0 // n->inf (1/n)=0

therefore, p(Xn=1)=0 as n goes to infinity

now, x1, x2, ....., xn are bernouli random variables that takes value 0 or 1

therefore, Xn takes 0 or 1

now, probability(Xn=1)=1/n and from above , we saw that as n goes to infinity this becomes 0

therefore, random variable Xn has 0 probability of becoming 1

hence, probability of becoming 0 is 1 // q=1-p=1-0=1

therefore, Xn goes to 0 as n tends to infinity