# Probability

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)=

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]=

Ty a lot

1. 👍 8
2. 👎 0
3. 👁 3,105
1. E= 10

10/(10-i)

29.29

1. 👍 5
2. 👎 1
2. @juanpro - E =10 - I am pretty confused.
Can you please explain how it is 10?

1. 👍 1
2. 👎 0
3. E(10) Must be 0. All Other Answers are Right though.

1. 👍 4
2. 👎 1
4. 𝑋=Number of tosses until getting 4 (including the last toss) by tossing a fair 10-sided die.
𝐄(𝑋)=

1. 👍 4
2. 👎 0

1. 👍 0
2. 👎 0
6. E(10)=0

The induction step is as follows. For i=1,…9:
E(i) = (E(i)+1)×i10+(E(i+1)+1)×(1−i10)
⟺ E(i) = E(i+1)+1010−i.

Using E(10)=0 and the induction step, we have
E(0) = 1010+109+…+102+101+0≈29.28968.

1. 👍 0
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7. 2) 10/(10-i)
sorry for the typo

1. 👍 0
2. 👎 0

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