Question

# 1) A 10-sided die is rolled infinitely many times. Let X be the number of rolls...

1) A 10-sided die is rolled infinitely many times. Let X be the number of rolls up to and including the first roll that comes up 2. What is Var(X)?

2) A 14-sided die is rolled infinitely many times. Let X be the sum of the first 75 rolls. What is Var(X)?

3) A 17-sided die is rolled infinitely many times. Let X be the average of the first 61 die rolls. What is Var(X)?

I know the answers but don't know how to get them. Please show work

1)

here this is geometric distribution with paramter p=1/10 (cause each number has equal probability)

Var(X) =(1-p)/p2 =(1-1/10)/(1/10)2 =100*9/10=90

2)

expected value on a single roll E(X)=(1+2+3+4+5+6+7+8+9+10+11+12+13+14)/14 =7.5

and E(X2)=(12+22+32+42+52+62+72+82+92+102+112+122+132+142)/14=72.5

therefore variance of single roll = E(X2)-(E(X))2 =16.25

for 75 rolls variance of sum =16.25*(75)=1218.75

3)

as above expected value E(X)=9

E(X2)=105

Var(X)=24

therfore variance of average of 61 die rolls=24/61=0.3934