Reference no: EM132314844

Probability Theory Assignment -

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are True or False and also give the reason in support of your answer.

(a) If A and B are any two events defined on a sample space S then P(A ∪ B) = P(S) always holds.

(b) Cumulative distribution function of a discrete random variable is always strictly increasing.

(c) If X is a discrete random variable with probability mass function (pmf)

then value of a will be 1.

(d) If X and Y are two independent random variables then P[X ≥ 2|Y > 1] < P[X ≥ 2].

(e) Suppose that you spin the dial shown in the figure so that it comes to rest at a random position.

The probability that the dial will land somewhere between 0 and 45 will be 1/4.

Q2. First check whether the following function is a valid density function? If it is a valid density then obtain its cumulative probability function F(x). If it is a valid density then finally calculate P(7 ≤ X ≤ 8) either using f(x) or F(x).

Q3. (a) The joint density function of random variables X and Y is given by

Are X and Y independent?

(b) A particular game is played where the contestant spins a wheel that can land on the number 1, 5, 30 with probabilities of 0.50, 0.45 and 0.05, respectively. The contestant pays INR5 to play the game and is awarded the amount of money indicated by the number where the spinner lands. Is this a fair game? [By fair, it is meant that the contestant should have an expected return equal to the price she pays to play the game.]

Q4. (a) Suppose two fair dice are tossed where each of the 36 possible outcomes is equally likely to occur. Knowing that the first die shows a 4, what is the probability that the sum of the two dice equals at least 7.

(b) Suppose that there are m students in a room. What is the probability that at least two of them have the same birthday? Assume that every day of the year is equally likely to be a birthday, and disregard leap years. That is, assume there are always 365 days to a year. [Hint: Attack the problem by first calculating probability of complement event and then use P(E) = 1 - P(E)^-].

Q5. The A taxi Cab Company has 12 Ambassadors and 8 Fiats. If 5 of these taxi cabs are in the workshop for repair and an Ambassador is as likely to be in for repair as a Fiat, what is the probability that (i) 3 of them are Ambassadors and 2 are Fiats, (ii) at least 3 of them are Ambassadors, and (iii) all 5 are of the same make?

Q6. (a) The probability that a player hits a target is 0.24. He fires 6 times. What is the probability of hitting the target exactly twice?

(b) What is the probability that 5^th success is obtained in 9^th trail if probability of success and failure do not vary from trial to trail.

Q7. (a) Metro trains in a certain city run every 9 minutes between 6.15 a.m. to 11.15 p.m. What is the probability that a commuter entering the station at a random time during this period will have to wait at least five minutes?

(b) Obtain mean and variance for the beta distribution whose density is given by

f(x) = 280x³/(1+x)⁹,  0 < x <∞

Q8. (a) A car manufacturer purchases car batteries from two different suppliers A and B. Suppose supplier A provides 60% of the batteries and supplier B provides the rest. If 6% of all batteries from supplier A are defective and 4% of the batteries from supplier B are defective. Determine the probability that a randomly selected battery is not defective.

(b) An item is produced by a machine in large numbers. The machine is known to produce 5% defectives. A quality control engineer is testing the items randomly. What is the probability that at least 5 items are examined in order to get 2 defectives?