# 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

### 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 tends to 1 in probability.

### Explanation

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

The convergence in probability is

Using Chebychev’s inequality

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

## Question number: 2

» Probability » Convergence » In Distribution

Appeared in Year: 2013

### 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

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

For ε > 0,

Hence

……… (1)

Note that

…. (2)

From

## Question number: 3

» Probability » Convergence » In Distribution

Appeared in Year: 2014

### Describe in Detail

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

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

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

For ε > 0,

Hence

……… (1)

Note that

…. (2)

From