Probability-Modes of Convergences of Sequences of Random Variables (ISS Statistics Paper I (New 2016 MCQ Pattern)): Questions 8 - 10 of 11

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Question number: 8

» Probability » Modes of Convergences of Sequences of Random Variables » In Distribution

MCQ▾

Question

Let Xn=1n for nN+ and let X=0. Let fn and f be the corresponding probability density functions and let Fn and F be the corresponding distribution functions. Then which of the following option is/are correct?

Choices

Choice (4) Response

a.

Fn(x)1 as n for all xR

b.

Fn(x){ as n

c.

Fn(x)F(x) as n for all x0

d.

None of the above

Question number: 9

» Probability » Modes of Convergences of Sequences of Random Variables » In Mean Square

MCQ▾

Question

Suppose that (X1,X2,) is a sequence of independent random variables with (X=n3)=1n2 , P(X=0)=11n2 ; nN+ , then the following option is correct?

Choices

Choice (4) Response

a.

XnX almost everywhere

b.

E(Xn)0 as n

c.

XnXasn in probability

d.

Question does not provide sufficient data or is vague

Question number: 10

» Probability » Modes of Convergences of Sequences of Random Variables » In Distribution

MCQ▾

Question

Consider the following statements and then choose the option in which the given statements are equivalent.

I) P(XnXasn)=0

II) P(XnXasn)=1

III) P( XnX >ϵforinfinitelymanynN+)=1 for every ϵQ+.

IV) P( XnX >ϵforinfinitelymanynN+)=0 for every ϵ>0.

V) P( XkX <ϵforsomekn)0asn for every ϵ>0.

Choices

Choice (4) Response

a.

II and IV

b.

I, III, V

c.

I, V

d.

Question does not provide sufficient data or is vague

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