Probability-Convergence (ISS (Statistical Services) 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 Equation tends to 1 in probability.

Explanation

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

The convergence in probability is

Equation

Using Chebychev’s inequality

Equation

Equation

Equation

Equation

Equation

Thus, sufficient condition is that Equation convergence in probability to 1 is that Equation .

Question number: 2

» Probability » Convergence » In Distribution

Appeared in Year: 2013

Essay Question▾

Describe in Detail

Show that convergence in probability implies convergence in distribution.

Explanation

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

Equation

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

Equation

For ε > 0,

Equation

Equation

Hence

Equation ……… (1)

Note that

Equation

Equation …. (2)

From

… (21 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

Equation

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

Equation

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

Equation

For ε > 0,

Equation

Equation

Hence

Equation ……… (1)

Note that

Equation

Equation …. (2)

From

… (42 more words) …

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