6. The probability of convergence of a sequence of independent random vari- ables is equal to 0 or 1. If the sequence {Xn) is iid, and not constant with probability 1, then P[Xn converges] = 0.
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![16. The probability of convergence of a sequence of independent random vari-
ables is equal to 0 or 1. If the sequence {Xn) is iid, and not constant with
probability 1, then
P[Xn converges] = 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F624629b5-d5ad-4038-ae2e-e5270cd30f7d%2F04381de4-78e2-456a-a2b7-3609ebfcd469%2Fr9dr93rl_processed.jpeg&w=3840&q=75)
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- 5. Let the random variable Y,, have a distribution that is b(n, p). Prove that (Y/n)(1-Y/n) converges in probability to p(1-p).Suppose that anx" converges for x = 5. Must it also converge for x = 4? What about x = -3?3-if ix-x| CIB, then we say {x} converges to x with rate convergence and write by x=x+B₂ true false O
- If 2n=1ªn converges conditionally then En= 1|an| diverges 'n = Select one: True False5.7. Let the random variable Y, have a distribution that is b(n, p). (a) Prove that Y,/n converges in probability to p. This result is one form of the weak law of large numbers. (b) Prove that 1 Y/n converges in probability to 1-p.y=x-x3/3+x5/15-..... Find ratio test for convergence
- Suppose {fn}n=1 converges uniformly on E. Show {fn=1 converges pointwise on E.Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.(-1)" . n2 + 5n +2 Determine whether > converges abso- (1+ 5)" lutely, converges conditionally, or diverges. n=1