Tutorial 1 - adf dfasdf fgfsdf fghdfgh dfhfdgh fgh fg PDF

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Math 4404: Probability and Statistics Tutorial 1 1.

A college mathematics department sends 8 to 12 professors to the annual meeting of the Mathematical Society, which lasts five days. The hotel at which the conference is held offers a bargain rate of a dollars per day per person if reservations are made 45 or more days in advance, but charges a cancellation fee of 2 a dollars per person. The department is not certain how many professors will go. However, from the past experience it is known that the probability of the attendance of i

professors is

1 5

for

i=8, 9, 10,11, ∧12 . If the regular rate of the hotel is 2 a dollars per day per person, should the department make any reservations so that the average cost will be minimum? If so, how many? 2.

Two players compete against each other in a game of chance where Player A

wins with probability

1 2 , and Player B wins with probability . Every time Player A loses he must pay Player 3 3 B $1, while every time Player B loses he must pay Player A $3. Each time the two play the game, the results are independent of any other game. If the two players repeat the games 10 times, what is the expected amount of Player A ’s winnings? 3.

A particular circuit works if all of its 10 component devices work. Each circuit is tested before leaving the factory. Each working circuit can be sold for k dollars, but each nonworking circuit is worthless and must be thrown away. Each circuit can be built with either ordinary devices or ultra-reliable devices. An ordinary device has a failure probability of q=0.1 , while an ultra-reliable device has a failure probability of

q , independently of any other device. However, each ordinary device costs $1, while 2

an ultra-reliable device costs $3. Let Ro and Ru denote the profit of a circuit when the circuit is built by ordinary and ultra-reliable devices, respectively. a) What are the PMFs of Ro and Ru ? Note that the calculation of profit should include the loss for a nonworking circuit. b) What are the expected values of Ro and Ru , E[ Ro ] and E[ Ru ¿ . c) Should you build your circuit with ordinary devices or ultra-reliable devices to maximize your profit? Note that your answer will depend on the value of k . 4.

A game is often played in carnivals and gambling houses is called chuck-a-luck, where a player bets on any number 1 through 6. Then three fair dice are tossed. If one, two, or all three land the same number as the player’s, then he or she receives one, two, or three times the original stake plus his or her original bet, respectively. Otherwise, the palyer loses his or her stake. Let X bet the net gain of the player per unit stake. First find the probability mass function of X ; then determine the expected amount that the player will lose per unit of stake.

5.

There are two possible causes for a breakdown of a machine. To check the first possibility would cost C1 dollars, and if that were the cause of the breakdown, the trouble could be repaired at a cost or R1 dollars. Similarly, there are costs C2 and R2 associated with the second possibility. Let p and 1− p denote, respectively, the probabilities that the breakdown is caused by the first and second possibilities. Under what conditions on p , Ci , and Ri , where i=1, 2 , should we check the first possible cause of breakdown and then the second, as opposed to reversing the checking order, so as to minimize the expected cost involved in returning the machine to working order?

6.

A word is selected at random from the following poem of Persian poet and mathematician Omar Khayyam, translated by English poet Edward Fitzgerald. Find the expected value of the length of the word. The moving finger writes and, having writ Moves on; nor all your Piety nor Wit

Shall lure it back to cancel half a line, Nor all your tears wash out a word of it. 7.

A couple (a women and a man) have the 95% desire to have at least one boy and at least one girl. Let X be the random variable representing the number of children of a couple. What is the minimum value of X that might fulfill the desire of a couple. Assume that the events that a child is a girl and a boy are equally likely and independent of the gender of other children in the family.

8.

The simplest error detection mechanism used in data communication is parity checking. Usually messages sent consist of characters, each character consisting of a number of bits (a bit is the smallest unit of information and is either 1 or 0). Assume that the number of bits in a character is 7. In parity checking, a 1 or 0 is appended to the end of each character at the transmitter to make the total number of 1’s even (and the parity checking mechanism is known as even parity). The receiver checks the number of 1’s in every character received, and if the number of 1’s is odd it signals an error. Suppose that each bit in a character is received correctly with probability 0.999, independently of other bits of the characters. i.

Find the probability that character is received in error, but the error is not detected by the parity check mechanism. ii. Find the probability that the parity check mechanism detects the error, if one or more bits are incorrectly received. iii. Suppose that a message consisting of six characters is transmitted. Find the probability that the message is erroneously received (at least one character is erroneously received), but none of the errors is detected by the parity check mechanism....