Question

Let Let X

1
​
,X
2
​
,…,X
n
​
be a random sample from the Beta(1,1) distribution. (a) What is another name for this distribution? (b) Find E[X
(n)
​
] and Var[X
(n)
​
] where X
(n)
​
=max(X
1
​
,X
2
​
,…,X
n
​
). (c) Find E[X
(n)
2
​
] 9. Let X be a Bernoulli random variable with parameter p. Derive the moment generating function of X. 10. Let Let X
1
​
and X
2
​
have joint pdf f
X
1
​
,X
2
​

​
(x
1
​
,x
2
​
)=8x
1
​
x
2
​
⋅I
(0,x
2
​
)
​
(x
1
​
)⋅I
(0,1)
​
(x
2
​
) Find the pdf of X
1
​
/X
2
​
.

Answer 1

Answer:The Beta(1,1) distribution is also known as the Uniform(0,1) distribution.

(b) Let X_1,X_2,...,X_n be a random sample from the Beta(1, 1) distribution. Then we know that each Xi ~ U(0, 1).

The probability density function of U(a,b), where a < b is:

f(x)= { 0 if x<a or x>b (b-a)^{-1} if a<=x<=b }

Therefore, f(x_i)=I[0≤xi≤11]=xi

Since all Xi's are independent and identically distributed as per above equations, P(X(n)<=t)= P(Xi <= t for all i = 1 to n) = t^n

Now we can write the CDF of X(n): F(t)=P(X(n)<=t)=(t^n)

Taking derivative with respect to t gives us the PDF: f(t)=d/dt(F(t))=nt^(n-1)

Using this formula for f(t), we can now calculate E[X(n)] and Var[X(n)]:

E[X(n)]= Integral{x * nt^(n-1)}dx over [0, 1] =Integral{x^2 * nt^(n-2)}dx over [0 , 1] =[x^3/(3nt^{n−2})] from [0 , 12 ] =(n/(3(n+2)))

Var[X(n)]=(∫[(x-E[x])²f(x)])from[01​]=(∫[(y/n)(ny-t)/(n+2)tⁿ⁻¹])from[01]=[(-nt^{-(n+2)})/((n+4)(+))]from[01]=(n^2/(12*(n+2)^2))

(c) To find E[X(n)^2], we use the formula for Var(X(n)) that we obtained in part (b):

Var(X(n)) = E[X(n)^2] - [E[X(n)]]^2

Substituting the values of E[X(n)] and Var(X(n)), we get:

E[X(n)^2] = Var(X(n)) + [E[X(n)]]^2 = n^ 22 / (3*(n+22))^21

Explanation: