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Chapter 7: Q.7.18 (page 364)

Let X be the length of the initial run in a random ordering of n ones and m zeros. That is, if the first k values are the same (either all ones or all zeros), then X Ú k. Find E[X].

Short Answer

Expert verified

The E[X}of the problem if X be the length of the initial run in a random ordering of n ones and m zeros E[L]=nm+1+mn+1.

Step by step solution

01

Given information

X= length of the initial run in a random ordering of n ones and m zeros

k-value are same

02

Solution

The solution will be shown below,

E[L]=E[L∣First vlaue is one]n(n+m)

+E[L∣First value is0]m(n+m)

Now if first value =1,

Length of run will be position of first 0when considering remaining n+m-1values, of which n-1are one's and mare zero's

⇒E[L]=n+mm+1nn+m+n+mn+1mn+m

⇒E[L]=nm+1+mn+1

03

Final answer

The E[X]of the given problem Will be E[L]=nm+1+mn+1.

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Most popular questions from this chapter

Consider the following dice game: A pair of dice is rolled. If the sum is7,then the game ends and you win 0.If the sum is not 7,then you have the option of either stopping the game and receiving an amount equal to that sum or starting over again. For each value ofi,i=2,...,12, find your expected return if you employ the strategy of stopping the first time that a value at least as large as i appears. What value ofileads to the largest expected return? Hint: Let Xidenote the return when you use the critical value i.To computeE[Xi], condition on the initial sum.

Use Table 7.2 to determine the distribution of∑i=1nXi when X1,…,Xnare independent and identically distributed exponential random variables, each having mean1/λ.

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Suppose that X1and X2 are independent random variables having a common mean μ. Suppose also that VarX1=σ12 and VarX2=σ22. The value of μ is unknown, and it is proposed that μ be estimated by a weighted average of X1 and X2. That is, λX1+(1-λ)X2 will be used as an estimate of μ for some appropriate value of λ. Which value of λ yields the estimate having the lowest possible variance? Explain why it is desirable to use this value ofλ.

Urn 1contains 5white and 6black balls, while urn 2contains 8white and 10black balls. Two balls are randomly selected from urn 1and are put into urn 2. If 3balls are then randomly selected from urn 2, compute the expected number of white balls in the trio.

Hint: LetXi = 1if the i th white ball initially in urn 1is one of the three selected, and let Xi = 0otherwise. Similarly, let Yi = 1if the i the white ball from urn 2is one of the three selected, and let Yi = 0otherwise. The number of white balls in the trio can now be written as∑15Xi+∑18Yi
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