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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,鈰�,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)=鈭�(1)xipin=(1)6

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

Ex1+x2+鈰�+xn=Ex1+Ex2+Ex2+鈰�+Exn16+16+16+16+鈰�=n6

Hence, Ex1,Ex2,Ex2,鈰�,Exn all are equal.

Hence, the statement has been proven.

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Most popular questions from this chapter

Equation (10.12)isonly an approximation (but usually satisfactory). Show, however, that if you keep the second order termsin,(10.10)then

role="math" localid="1664364127028" w炉=w(x炉,y炉)+12(鈭�2w鈭�x2)蟽x2+12(鈭�2w鈭�y2)蟽y2.

Two cards are drawn at random from a shuffled deck.

  1. What is the probability that at least one is a heart?

(b) If you know that at least one is a heart, what is the probability that both are

hearts?

Define s by the equation.s2=(1/n)鈭�i=1n(xi鈭�x炉)2Show that the expected valueof.s2is[(n鈭�1)/n]蟽2Hints: Write

(xi鈭�x炉)2=[(xi鈭�渭)鈭�(x炉鈭�渭)]2=(xi鈭�渭)2鈭�2(xi鈭�渭)(x炉鈭�渭)+(x炉鈭�渭)2

Find the average value of the first term from the definition of蟽2and the average value of the third term from Problem 2. To find the average value of the middle term write

(x炉鈭�渭)=(x1+x2+鈰�+xnn鈭�渭)=1n[(x1鈭�渭)+(x2鈭�渭)+鈰�+(xn鈭�渭)]

Show by Problemthat

E[(xi鈭�渭)(xj鈭�渭)]=E(xi鈭�渭)E(xj鈭�渭)=0鈥夆赌夆赌塮辞谤i鈮�j

andevaluate6.14 (same as the first term). Collect terms to find

E(s2)=n鈭�1n蟽2

Suppose that Martian dice are 4-sided (tetrahedra) with points labeled 1鈭�4. When a pair of these dice is tossed, let x be the product of the two numbers at the tops of the dice if the product is odd; otherwisex=0.

Some transistors of two different kinds (call them N and P) are stored in two boxes. You know that there are 6 N鈥檚 in one box and that 2 N鈥檚 and 3 P鈥檚 got mixed in the other box, but you don鈥檛 know which box is which. You select a box and a transistorfrom it at random and find that it is an N; what is the probability that it came from the box with the 6 N鈥檚? From the other box? If another transistor is picked from the same box as the first, what is the probability that it is also an N?
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