There are many, hundreds, of density functions. But all of them can be studied using the concepts and methods that we have studied in this course. Take for example, the Beta distribution for a continuous random variable,
X
, that can only take values between 0 and 1 in the real line. The density function is
f(x)= B(α,β)1
​
x α−1 (1−x) β−1
,0≤x≤1.
and the expected value and variance of
X
are given by the following formulas
μ X​ = α+βα
​
,σ X2​ = (α+β) 2 (α+β+1)αβ
​
.
Suppose
α=37,β=463
. The
B(α,β)
is just the mathematical beta function, an integral with special meaning to mathematicians.
B(α,β)=∫ 01 u α−1 (1−u) β−1 du
Answer the following questions showing work and justifying your answers (that is, if you use results or theorems proved or stated in class, say which). (a) Consider
X 1
​
,X 2
​
,……,X 120
​
independent identically distributed (iid) random variables (what statisticians call a random sample), all of which follow the preceding Beta pdf with those given parameter values. Calculate the probability that the average of those random variables is larger than
0.06
. (b) Consider again
X 1
​
,X 2
​
,……,X 120
​
of part (a). Calculate the joint probability that each and everyone of these random variables is less than
0.06
. You may use the app for the Beta distribution as needed. (c) Consider again
X 1
​
,X 2
​
,……,X 120
​
of part (a) and (b). What would be the probability that in 40 out of these 120 random variables the value is less than
0.06
? (c) Focus now just in one of the random variables,
X
. Suppose all we know is the expected value of
X
and the variance of
X
, as given earlier at the beginning of this problem. But suppose that we do not know that
X
is a Beta distribution. Can you find the shortest possible interval such that the probability that
X
is in that interval is at least
0.9
? (d) Compare the
0.9
probability of part (c) with the probability of the interval that you obtain in part (c) using the Beta distribution given in this problem. That is, say your interval is
(r,s)
. What is the probability that
X
with that Beta distribution is in that interval? That would be the exact probability for that interval. You may use the app for the Beta distribution
Θ −
​
, but you must explain how you are using the app to calculate the probability.