Consider a random sample ?1, ?2, … , ?? from a population distributed with the following mass function:

Consider a random sample ?1, ?2, … , ?? from a population distributed with the
following mass function:
?(?) = {
3??−3
?! , ??? ? = 0,1, … . , ∞
0, ?????ℎ???
a) What is the mean and variance of a statistic ? = ∑ ??
10
?=1 ?
b) What is the mean of the statistic ? = ∏ ??
10
10
?=1 ?
c) Find the probability distribution function of ? = ∑ ??
10
?=1 .
d) What is the probability that ? = ∑ ??
10
?=1 is at least 3?
2) Let ?1, ?2, ?3, ?4, ?5 be an independent and identically random variables following a
Binomial distribution with 1
4 probability of success and 5 independent trials.
a) What is the expected value of ? = ∏ (??2 + ??)5
?=1 ?
b) Find the probability distribution function of ? = ∑ ??
3
?=1 .
c) Find the mean and variance of ? = ∑ ??
3
?=1 .
d) What is the ?(0 < ? ≤ 3)?
3) Suppose that ?1, ?2, … , ?100 is a random sample from a normally distributed
population with mean ? = 25 and variance ?2 = 36. If ? = 1
100 ∑ ??
100
?=1 , ?2 =
1
99 ∑ (?? − ?)2100
?=1 , and ? = ?−??
??
a) Find the mean and variance of ?.
b) Find the ?(1000 < ∑ ??
100
?=1 < 1200).
c) State with parameters the probability distribution function of ?.
d) Compute the ?(0.004 < ?2 < 3.841).
e) Compute the ?(99?2 > 2805.444)