S² P, o² and „", → 1, their product converges in *6. In lecture it is claimed that since "S2 probability to o²; in other words, that the product of a convergent sequence of real numbers and a sequence of random variables with a limit in probability converges in probability to the product of the limits. Suppose that {Yn} is a sequence of random variables that converges in probability to Y: lim P (|Yn – Y|> e) = 0 Ve>0 Let {an} be a sequence of real numbers that converges to a. That is, for any e > 0, there is some n* such that n > n* Jan – al < e Show that anYn 24 aY. (Hint: show that an → a and use the property (without proof) that convergence in probability is preserved under multiplication.)
S² P, o² and „", → 1, their product converges in *6. In lecture it is claimed that since "S2 probability to o²; in other words, that the product of a convergent sequence of real numbers and a sequence of random variables with a limit in probability converges in probability to the product of the limits. Suppose that {Yn} is a sequence of random variables that converges in probability to Y: lim P (|Yn – Y|> e) = 0 Ve>0 Let {an} be a sequence of real numbers that converges to a. That is, for any e > 0, there is some n* such that n > n* Jan – al < e Show that anYn 24 aY. (Hint: show that an → a and use the property (without proof) that convergence in probability is preserved under multiplication.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![*6. In lecture it is claimed that since "-1S² 2→ o² and „", → 1, their product converges in
probability to o²; in other words, that the product of a convergent sequence of real numbers
and a sequence of random variables with a limit in probability converges in probability to the
product of the limits.
Suppose that {Yn} is a sequence of random variables that converges in probability to Y:
lim P (|Y, – Y| > e) = 0
Ve > 0
n-00
Let {an} be a sequence of real numbers that converges to a. That is, for any e > 0, there is
some n* such that
n > n*
|an – al < e
Show that a, Yn 2» aY. (Hint: show that an » a and use the property (without proof)
that convergence in probability is preserved under multiplication.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F544a9f38-3e68-4113-87a7-993914a696cb%2F188fb26a-c8cf-45cf-b0b8-512efc1bf918%2Fiwdzk8_processed.png&w=3840&q=75)
Transcribed Image Text:*6. In lecture it is claimed that since "-1S² 2→ o² and „", → 1, their product converges in
probability to o²; in other words, that the product of a convergent sequence of real numbers
and a sequence of random variables with a limit in probability converges in probability to the
product of the limits.
Suppose that {Yn} is a sequence of random variables that converges in probability to Y:
lim P (|Y, – Y| > e) = 0
Ve > 0
n-00
Let {an} be a sequence of real numbers that converges to a. That is, for any e > 0, there is
some n* such that
n > n*
|an – al < e
Show that a, Yn 2» aY. (Hint: show that an » a and use the property (without proof)
that convergence in probability is preserved under multiplication.)
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