Question

Expectation 1

Compute ?(?) for the following random variable ? :
?=Number of tosses until getting 4 (including the last toss) by tossing a fair 10-sided die.         
?(?)= ?

-------------------------------------------------------------------------------------------
Expectation 2

Compute ?(?) for the following random variable ? :
?=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 ?(?) be the expected number of additional tosses until all 10 numbers are seen (including the last toss) given ? distinct numbers have already been seen.

1.Find ?(10) .

?(10)= ?

---------

2. Write down a relation between ?(?) and ?(?+1) . Answer by finding the function ?(?) in the formula below.

For ?=0,1,…,9 :

    ?(?)=?(?+1)+?(?)         
where ?(?)=
  
---------

3. Finally, using the results above, find ?[?] .

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

?[?]=

-------------------------------------------------------------------------------------------

Answer 1

1.

Let X be the distribution of number of tosses until getting a 4 for a fair 10-sided die.

We have to find the probability of getting 4 by tossing a fair 10-sided die.

Thus, we can say X follows a geometric distribution such that

Thus,

Thus, we can write,

Then the expected value for X where X is the number of trials up to and including the first success is given by:

2.

We can define E(10) as the expected number of tosses until all 10 numbers are seen given 10 distinct numbers have already been seen.

Thus, we can write

Let us define f(i) as the difference of expected number of additional tosses until all 10 numbers are seen given i distinct numbers have already been seen and the expected number of additional tosses until all 10 numbers are seen given i+1 distinct numbers have already been seen.

This can be written as the 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, then,

Remaining distinct numbers =

P (next distinct number) =

Let X be the number of trails to get the next (i+1)th distinct number.

Then, X follows a geometric distribution. It can be written as:

Thus, we can write

and so on,

which is the required expected value.