Questions on Probability and Random Process

7/21/2019 Questions on Probability and Random Process

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Assignment- 1

RANDOM PROCESS AND APPLICATIONS

1. a) State Bernoullis theorem on independent trials.

b) A fair dice is rolled 5 times. Find the probability that 1 shows twice, 3

shows twice and 6 shows once.

2. In the ternary communication channel shown in figure, a 3 is sent three times

more frequently than a 1and a 2is sent two times more frequently than a 1. A

1is observed, what is the conditional probability that a1was sent?

3. A card is selected at random from a standard deck of 52 cards. Let A be the

event of selecting an ace and let B be the event of selecting a red card;.

There are 4 aces and 26 red cards in the normal deck. Are A and B

independent?

4. Assume there are three machines A, B and C in a semiconductor

manufacturing factory that makes chips. They manufacture respectively 25,

35, 40 percent of total semiconductor chips. Of their outputs, respectively 5, 4

and 2 percent of the chips are defective. A chip is drawn randomly for m the

combined output of the three machines and it is found defective. What is the

probability that the defective chip was manufactured by machine A? (ii) By

machine B? (iii) By machine C?

5. An automatic breathing apparatus used in anesthesia fails with probability pB .

A failure means death to the patient unless a monitoring system(M) detects

the failure and alerts the physician. The monitor system fails with probability

pM . The failure of the systems is independent events. Professor X argues that

if pM> pB, installation of M is useless. Show that Prof X needs to have a

course on probability theory by comparing the probability of a patient dyingwith and without the monitor system in p[lace. Take pM=0.1=2pB.

6. In a restaurant known for its unusual service , the time X in minutes that a

customer has to wait before he capture the attention of a waiter is specified

by the following distribution function:


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(a)Sketch (b) Compute and sketch pdf . Verify that area under pdfis unity. (c) What is the probability that customer will have to wait (i) atleast 10

minutes (ii) less than 5 minutes (iii) between 5 and 10 minutes (iv) exactly one

minute.

7. The pdf of a random variable X is shown in figure. The numbers in

parenthesis indicate area.(a)Compute the value of A (b) Sketch the PDF (c) Compute (d)Compute (d) Find

8. Write the probability density functions (using Dirac Delta functions) for the

Bernoulli, binomial and Poisson PMFs.

9. The time to failure in months X of light bulbs produced at two manufacturing

plants A and B obey respectively the following PDFs:


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( ) and ( ) Plant B produces 3 times as many bulbs as plant A. the bulbs

indistinguishable to the eye are intermingled are sold. What is the probability

that bulb purchased at random will burn atleast (a) 2 months (b) 5 monthsand (c) 7 months.

10. A US defense radar scans the skies for UFOs. Let M be the event that a UFO

is present and Mcbe the event that a UFO is absent. Let ||

be the conditional pdf of radar return signal X when UFO isactually there an d let||

be the conditional pdf of radarreturn signal X when there is no UFO. To be specific let and let the alertlevel be

Let A denote the event of an alert that is

* +. Compute

||,|and ,|.

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Questions on Probability and Random Process