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QUESTION: 1

Given X ¬ B (n,p) if E(X) = 6, Var(X) = 4.2 then what is the value of n?

Solution:

QUESTION: 2

The probability density function of the normal distribution of a random variable X is

What is the sum of mean and standard deviation?

Solution:

Probability density function of normal distribution is

Note:

σ = standard deviation

σ^{2 }= variance

μ = mean

QUESTION: 3

Binomial distribution B(n,p) can be approximated to a normal distribution N(np, np(1−p)) if ____

Solution:

Binomial distribution B(n,p) can be approximated to normal distribution N(np,np(1−p)) if n is large and p and 1-p are almost equal. Approximation generally improves as *n* increases (at least 20) and is better when *p* is not near to 0 or 1.

*Answer can only contain numeric values

QUESTION: 4

If a random variable x satisfies the Poisson’s distribution with a mean value of 3, then the probability that (x ≥ 2) is Poisson’s distribution,

Solution:

For Poisson’s distribution,

QUESTION: 5

Let Harsh and Dinesh be the two players playing chess and their chances of winning a game are in the ration 4:3 respectively. What is the chance of Dinesh winning at least 4 games out of five games played?

Solution:

*Answer can only contain numeric values

QUESTION: 6

From an urn containing 3 red and 2 white balls, a man is to draw 2 balls at random without replacement, being promised Rs. 20 for each red ball he draws and Rs. 10 for each white one. Find his expectation (In rupees).

Solution:

A man is to draw 2 balls at random without replacement.

Probability to draw 2 red balls

Probability to draw 2 white balls

Probability to draw one red ball and one white ball =

Let X be the random variable and it shows the money he earn.

Expectation =

*Answer can only contain numeric values

QUESTION: 7

If the probability of passing an exam is 0.001, then determine the chance that more than 3 students out of 3,000 will pass the exam (answer up to 2 decimal place)?

Solution:

Poisson distribution is used as probability of occurrence is very small.

mean = λ = n

∴ λ = 3000 × 0.001 = 3

= 0.35

QUESTION: 8

A dice is rolled 180180 times, find the probability that face 44 will turn up at least 3535 times.(Assume normal distribution). Assume p(0 < z < 1) = 0.3413

Solution:

Given that n = 180

Required Probability is given by

∴ mean

Variance = npq = 25

= 0.5 – 0.3413 = 0.1587

*Answer can only contain numeric values

QUESTION: 9

The probability that GATE CS/IT question has an error is 1/20 and 65 questions are made in such an exam. If the probability that at least 2 question has an error is [1−[a(19/20)^{64}]] then what is the value of **a **(answer up to 2 decimal place)**?**

Solution:

Using Binomial Distribution

Probability that x questions have an error

QUESTION: 10

If X is a Poisson variate such that P(X=2) = 3P(X=4) then what is the value of P(X=3)?

Solution:

P (X = 2) = 3P (X = 4)

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