Exercise: Convergence in probability

Question

a) Suppose that Xn is an exponential random variable with parameter λ=n. Does the sequence {Xn} converge in probability?

b) Suppose that Xn is an exponential random variable with parameter λ=1/n. Does the sequence {Xn} converge in probability?

c) Suppose that the random variables in the sequence {Xn} are independent, and that the sequence converges to some number a, in probability. Let {Yn} be another sequence of random variables that are dependent, but where each Yn has the same distribution (CDF) as Xn. Is it necessarily true that the sequence {Yn} converges to a in probability?

Answer 1

a) Yes
b) No
c) Yes