Question

Expectation 1

Compute E(X) for the following random variable X

: X=Number of tosses until getting 4 (including the last toss) by tossing a fair 10-sided die.

E(X)= ?

Expectation 2

Compute E(X) for the following random variable X : X=Number of tosses until all 10 numbers are seen (including the last toss) by tossing a fair 10-sided die.

To answer this, we will use induction and follow the steps below:

Let E(i) be the expected number of additional tosses until all 10 numbers are seen (including the last toss) given i distinct numbers have already been seen.

Find E(10) .

E(10)= ?

Write down a relation between E(i) and E(i+1) .

Answer by finding the function f(i) in the formula below. For i=0,1,…,9 : E(i)=E(i+1)+f(i)

where f(i)= ?

Finally, using the results above, find E[X] .

(Enter an answer accurate to at least 1 decimal place.)

E[X]=?

Answer 1

Expectation 1

The probability of getting 4 by tossing a fair 10-sided die = 1/10

X ~ Geom(p = 1/10)

E(X)= 1/p = 1/(1/10) = 10

Expectation 2

E(10) = Expected number of additional tosses until all 10 numbers are seen given 10 distinct numbers have already been seen

= 0

f(i) = E(i) - E(i+1) = Expected number of additional tosses until all 10 numbers are seen given i distinct numbers have already been seen - Expected number of additional tosses until all 10 numbers are seen given i+1 distinct numbers have already been seen

= Expected number of additional tosses to get the next (i+1)th distinct number given i distinct numbers have already been seen

When i distinct numbers have already been seen, remaining distinct numbers = 10 - i

Probability to find next distinct number = (10 - i) / 10

Let X be the number of trails to get the next (i+1)th distinct number. Then X ~ Geom(p = (10 - i) / 10)

E(X) = 1/p = 10/(10-i)

Thus,

f(i) = Expected number of additional tosses to get the next (i+1)th distinct number given i distinct numbers have already been seen = 10/(10-i)

E[X] = E[0] = E[1] + f(0)

= E[2] + f(1) + f(0)

= E[3] + f(2) + f(1) + f(0)

.....

= E[10] + f(9) + .... + f(1) + f(0)

= 0 + f(9) + f(8) + ... + f(1) + f(0)