Assighnment.docx

ASSIGNMENT GROUP-1 • The life in hours of a certain kind of radio tube has the pdf f(x)=

Find (i) the distribution function F(x) (ii) the probability that the life of tube is 150 hours. •

 Let take a customer who goes for shopping and lets suppose the time he spends in the shop is exponentially distributed with a mean value 15 minutes. Then what will be the probability that the customer will spend more than 20 minutes in shopping? What is the probability that the customer will spend more than 20 minutes in the bank given that he is still in the shop after 15 minutes?

GROUP-2 • (a)Four coins are tossed. What is the expectation of the number of heads. • (b) The diameter of an electric cable is assumed to be a continous virate with p.d.f f(x) = 6(x)(1-x), 0 x 1. Verify that the above is p.d.f. Also, find the mean and variance.

• At an urgent care facility, patients arrive at an average rate of one patient every seven minutes. Assume that the duration between arrivals is exponentially distributed. (a)Find the probability that the time between two successive visits to the urgent care facility is less than 2 minutes.

GROUP-3 • Determine the constant ‘C’ so that the following function can serve as the probability distribution of a discrete random variable with the given range f(x) = cs, x = 1, 2, 3, 4, 5. • Define CDF and PDF of a continuous random variable. How does the PDF of a continous random variable differ from pmf of a discrete variable? Find k so that

, zero, elsewhere, is a proper

pdf of a continuous probability distribution. • Evaluate (i) expected (or mean) (ii) Variance and (iii) SD and (iv) MGF for the following probability distributions (a) Distribution Bernaulli (b) Continuous Uniform Distribution. (c) exponential Distribution

GROUP-4 • Define a MGF of a random variable. Evaluate (i) expected (or mean) (ii) Variance and (iii) SD and (iv) MGF for the following probability distributions (a) Geometric Distribution (b) Gamma Distribution. (c) beta distribution.

• Find the expected value of the product of points on n dice • X denotes the profit that a man can make in business. He may earn Rs. 2800 with probability ½, he may lose Rs. 5000 with probability 3/10 and he may neither lose nor gain with probability 1/5. Show that the mathematical expectation is 100

GROUP-5 • The time X(in years) required to complete a software project has a pdf of the

form

f(x)=kx(1-x) for 0<x<1 and zero elsewhere. (i) Find k (ii) Find the probability

that

the

project will be completed in less than four months. • Explain a Gamma Distribution. Find its Mean value, Variance and Coefficient of Variation. • A petrol pump is supplied with petrol once a day. If its daily volume of sales (X) in thousands of litres is distributed by: f(x) = 5(1-x)4, 0≤x≤1.what must be the capacity of its tank in order that the probability that its supply will be exhausted in a given day shall be 0.01?

GROUP-6 • A random variable x has the following probability function; Value of x : P(x)

:

-2

-1

0

1

2

0.1

k

o.2

2k

0.3

Find the value of k and calculate mean and variance.

k

 An urn contains 4 white and 3 red balls. Three balls are drawn, with replacement, from this urn. Find 𝜇, 𝜎 𝑎𝑛𝑑 𝜎 2 for the number of red balls drawn.  If a r.v. X has the MGF M(t) =

3 3−𝑡

, obtain the standard deviation

of X.

GROUP-7 • Define cdf (Commulative distribution function) and pdf of a continuous random variable. Find k so that , zero, elsewhere, is a proper pdf of a continuous probability distribution. • In an amusement fair, a competitor is entitled for a prize if he throws a ring on a peg from a certain distance. It is observed that only 30% of the competitors are able to do this. If someone is given 5 chances, what is the probability of his winning the prize when he has already missed 4 chances?