Question

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)?

Answer: 90.0

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

Answer: 1218.75

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)?

Answer: 0.3934

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

Answer 1

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