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

Essay Question▾

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

Equation

Equation

Using Chebyshev’s inequality,

Equation

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

Equation

Equation

So, the number of heads comes between

… (34 more words) …

Question number: 2

» Probability » Tchebycheffs Inequality

Appeared in Year: 2013

Essay Question▾

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

Equation

or Equation

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Equation

Using Chebyshev’s inequality

… (2 more words) …

Question number: 3

» Probability » Tchebycheffs Inequality

Appeared in Year: 2011

Essay Question▾

Describe in Detail

Let X be a positive valued random variable. Prove that

Equation

Hence deduce the Chebychev’s inequality.

Explanation

The expectation of X is define as

Equation

For any x ≥ r

Equation

Equation

Equation

This implies that

Equation

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

Equation

Question number: 4

» Probability » Tchebycheffs Inequality

Appeared in Year: 2015

Essay Question▾

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

Equation

or Equation

σ 2 = E [X 2] - (E [X] ) 2 =4

Then, a lower bound for the probability

Equation

Using Chebyshev’s inequality

… (2 more words) …

Question number: 5

» Probability » Tchebycheffs Inequality

Appeared in Year: 2009

Essay Question▾

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

Equation

Let X be a random variable with meanE (X) = µ and variance Var (X) = σ 2. Then any

… (11 more words) …

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