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Chapter 15: Q14P (page 750)

As in Problem 11, show that the expected number of5's in n tosses of a die is n6.

Short Answer

Expert verified

Ex1,Ex2,Ex2,⋯,Exnall are equal.

The statement has been proven.

Step by step solution

01

Given Information

A coin is tossed twice.

02

Definition of the cumulative distribution function.

The likelihood that a comparable continuous random variable has a value less than or equal to the function's argument is the value of the function.

03

Prove the statement

A dice is tossed, the outcomes isS={1,2,3,4,5,6} .

E(x)=∑(1)xipin=(1)6

If the dice are tossed n times, then the expectation is given below.

Ex1+x2+⋯+xn=Ex1+Ex2+Ex2+⋯+Exn16+16+16+16+⋯=n6

Hence, Ex1,Ex2,Ex2,⋯,Exn all are equal.

Hence, the statement has been proven.

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Most popular questions from this chapter

Show that if w=xyor,w=x/ythen(10.14) givesthe convenient formula for relative error

rww=(rxx)2+(ryy)2

Two decks of cards are “matched,” that is, the order of the cards in the decks is compared by turning the cards over one by one from the two decks simultaneously; a “match” means that the two cards are identical. Show that the probability of at least one match is nearly.1−1/e

A letter is selected at random from the alphabet. What is the probability that it is one of the letters in the word “probability?” What is the probability that it occurs in the first half of the alphabet? What is the probability that it is a letter after x?

(a) Set up a sample space for the 5 black and 10 white balls in a box discussed above assuming the first ball is not replaced. Suggestions: Number the balls, say 1 to 5 for black and 6 to 15 for white. Then the sample points form an array something like (2.4), but the point 3,3 for example is not allowed. (Why?

What other points are not allowed?) You might find it helpful to write the

numbers for black balls and the numbers for white balls in different colors.

(b) Let A be the event “first ball is white” and B be the event “second ball is

black.” Circle the region of your sample space containing points favorable to

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number of sample points in A and in B; these are and . The region

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. Use the numbers you have found to verify (3.2) and (3.1). Also find

and and verify (3.3) numerically.

(c) Use Figure 3.1 and the ideas of part (b) to prove (3.3) in general.

Suppose you have 3 nickels and 4 dimes in your right pocket and 2 nickels and a quarter in your left pocket. You pick a pocket at random and from it select a coin at random. If it is a nickel, what is the probability that it came from your right pocket?
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