# Probability-Tchebycheffs Inequality (ISS (Statistical Services) Statistics Paper I (Old Subjective Pattern)): Questions 1 - 5 of 5

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## Question number: 1

» Probability » Tchebycheffs Inequality

Appeared in Year: 2014

### Describe in Detail

Show that for 40, 000 throws of a balanced coin, the probability is at least 0.99 that the proportion of heads will fall between 0.475 and 0.525.

### Explanation

For balanced coin the probability is p = 0.5. For Bernoulli trails where n = 40, 000, the mean and standard deviation is

Using Chebyshev’s inequality,

The question says that the probability is at least 0.99 that is

So, the number of heads comes between… (34 more words) …

## Question number: 2

» Probability » Tchebycheffs Inequality

Appeared in Year: 2013

### Describe in Detail

Let X be a random variable with E [X] = 4 and E [X ^{2}] = 20. Use Chebyshev’s inequality to determine a lower bound for the probability P [0 < x < 8].

### Explanation

Let X be a random variable with mean µ and variance σ ^{2}. Then any k > 0, the Chebyshev’s inequality is

or

σ ^{2} = E [X ^{2}] - (E [X] ) ^{2} =4

Then, a lower bound for the probability

Using Chebyshev’s inequality… (2 more words) …

## Question number: 3

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

### Describe in Detail

Let X be a positive valued random variable. Prove that

Hence deduce the Chebychev’s inequality.

### Explanation

The expectation of X is define as

For any x ≥ r

This implies that

Hence, any function of X, assume g (x) the Chebychev’s inequality is

## Question number: 4

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

### Describe in Detail

Let X be a random variable with E [X] = 3 and E [X ^{2}] = 13. Use Chebyshev’s inequality to obtain P [-2 < X < 8].

### Explanation

Let X be a random variable with mean µ and variance σ ^{2}. Then any k > 0, the Chebyshev’s inequality is

or

σ ^{2} = E [X ^{2}] - (E [X] ) ^{2} =4

Then, a lower bound for the probability

Using Chebyshev’s inequality… (2 more words) …

## Question number: 5

» Probability » Tchebycheffs Inequality

Appeared in Year: 2009

### Describe in Detail

Let X ~ BIN (100, 0.2). Compute P [10 ≤ X ≤ 30].

### Explanation

X ~ BIN (100, 0.2), where n = 100, p = 0.2, q = 0.8

E (X) =np = 20, Var (X) =npq = 16, σ = 4

Let X be a random variable with meanE (X) = µ and variance Var (X) = σ ^{2}. Then any… (11 more words) …