qwertykeyboards

05-19-2010, 05:44 AM

Hi,

the following question keeps coming up in recent year's exam and i'm unsure how to do it fully:

Q. If the sequence of random variables {Xn}n=1 up to infinity, is such that, for each n element of N,

n with probability 1/n

Xn =

0 with probability [1-(1/n)]

answer the following, justifying your answer in each case:

(i) Does Xn converge in probability?

(ii) Does Xn converge in L1?

i have done a similar problem and got it right but don't fully know the answer to this one. for (i) i have: when Xn converges to X in prob. lim n->infinity P (mod of Xn-X < epsilon) = 1, for all epsilon > 0. so plugging in Xn = n would give me 0< epsilon, meaning prob of this would have to be 1 since epsilon is greater than 0. the same for Xn = 0. my problem with this is that it doesnt involve the given probabilities 1/n and 1-1/n so i feel theres more to it.

same thing with (ii), where do the probs come into it? convergence in L1 is lim of E[ mod Xn-X] = 0. plugging in Xn = n and Xn = 0 both give 0 so id say that Xn does converge in both probability and in L1.

am i correct? i'm sure they didnt just put the probabilities in the question just for show so i must be wrong.

any help would be greatly appreciated!!! exam tomorrow.

the following question keeps coming up in recent year's exam and i'm unsure how to do it fully:

Q. If the sequence of random variables {Xn}n=1 up to infinity, is such that, for each n element of N,

n with probability 1/n

Xn =

0 with probability [1-(1/n)]

answer the following, justifying your answer in each case:

(i) Does Xn converge in probability?

(ii) Does Xn converge in L1?

i have done a similar problem and got it right but don't fully know the answer to this one. for (i) i have: when Xn converges to X in prob. lim n->infinity P (mod of Xn-X < epsilon) = 1, for all epsilon > 0. so plugging in Xn = n would give me 0< epsilon, meaning prob of this would have to be 1 since epsilon is greater than 0. the same for Xn = 0. my problem with this is that it doesnt involve the given probabilities 1/n and 1-1/n so i feel theres more to it.

same thing with (ii), where do the probs come into it? convergence in L1 is lim of E[ mod Xn-X] = 0. plugging in Xn = n and Xn = 0 both give 0 so id say that Xn does converge in both probability and in L1.

am i correct? i'm sure they didnt just put the probabilities in the question just for show so i must be wrong.

any help would be greatly appreciated!!! exam tomorrow.