Q.7.16 - A First Course in Probability

Question

In Self-Test Problem $7.1$, we showed how to use the value of a uniform ($0,1$) random variable (commonly called a random number) to obtain the value of a random variable whose mean is equal to the expected number of distinct names on a list. However, its use required that one choose a random position and then determine the number of times that the name in that position appears on the list. Another approach, which can be more efficient when there is a large amount of replication of names, is as follows: As before, start by choosing the random variable X as in Problem $7.1$. Now identify the name in position X, and then go through the list, starting at the beginning, until that name appears. Let I equal $0$if you encounter that name before getting to position X, and let l equal1ifyour first encounter with the name is at position X. Show that E[mI]=d.

 Hint: Compute E[I] by using conditional expectation.

Step-by-Step Solution

Answer

The position is equality likely Therefore, to be chosen as,

$E\left[mI\right]=E[m\mid n(X\left)\right]=d$

Step 1: Given Information

Show that $\mathrm{E}\left[\mathrm{ml}\right]=\mathrm{d}.$

Step 2: Explanation

Name chosen as n(x),

position is equality likely Therefore, to be chosen as

$E[I\mid n(X\left)\right]=P\{I=1\mid n(X\left)\right\}=\frac{1}{N\left(X\right)}$

$E\left[I\right]=E[I\mid n(X\left)\right]$

Therefore,

$E\left[mI\right]=E[m\mid n(X\left)\right]=d$

Step 3: Final Answer

$E\left[mI\right]=E[m\mid n(X\left)\right]=d$