Probability-Convergence (ISS Statistics Paper I (Old Subjective Pattern)): Questions 1 - 3 of 3

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Question number: 1

» Probability » Convergence » In Probability

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X 1, X 2, …, X n be a sequence of i. i. d. r. v. s with E (X i) = 0 and V (X i) = 1. Show that the sequence Sn2=1ni=1nXi2 tends to 1 in probability.

Explanation

we have known that Sn2 is the sample variance of the sequence. The mean is

The convergence in probability is

P[XnX<ϵ]=1asn

Using Chebychev’s inequality

P[Sn2E(S… (177 more words) …

Question number: 2

» Probability » Convergence » In Distribution

Appeared in Year: 2013

Essay Question▾

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Show that convergence in probability implies convergence in distribution.

Explanation

The sequence X n converges to X in probability if for any ε > 0

P(XnXε)0asn

The sequence X n converges to the distribution of X as n tends to infinity if

Fn(… (303 more words) …

Question number: 3

» Probability » Convergence » In Distribution

Appeared in Year: 2014

Essay Question▾

Describe in Detail

{X n} is a sequence of independent variables. Show that

XnPXXnDX

where X is a random variable. Is the converse true?

Explanation

The sequence X n converges to X in probability if for any ε > 0

P(XnXε)0asn

The sequence X n converges to the distribution of X as n tends to infinity if

Fn(… (314 more words) …

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