# 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

… (116 more words) …

## 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,

… (231 more words) …

## 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,

… (240 more words) …