Question

Please write in detail neatly. Kindly don't use any symbols and shortcut words. Please write the formula appropriately. The question is:

Consider an unbiased 4-sided die where the sides are numbered 1, 2, 3, 4, and a biased coin with probability of head P(H) =4 divided by 7. A chance experiment consists of rolling the die once and then tossing the coins many times as the number showing on the die. Let X represent the outcome of the roll of the die, and let Y represent the number of heads observed after tossing the coin. Below, find each of the following:

a. R(X) and R(Y)

b. Give the distribution of X. What is the name of this distribution?

c. What is the name of the conditional distribution P(Y | X = 2). Give this distribution.

d. Give the conditional expectation E[Y | X = 2].

e. Give the joint distribution of the random vector < X, Y >.

f. Give P(Y = 3).

g. Give P(X = 2 | Y = 2).

h. Give E[E[Y|X]].

Please explain to me about "unbiased 4-sided die" and "a biased coin." I just need to understand the meaning of them. Could you please give two examples so that I can understand easily?

Answer 1

X= Outcome of roll of a die

{1,2,3,4}

P(X=x)= Prob that (X=x)

P(X=1)=P(X=2)=P(X=3)=P(X=4)=1/4

R(x)=P(X<= x)

R(1)=P(X<=1) = P(X=1)=1/4

R(2)=P(X<=2)=P(X=1)+P(X=2) = (1/4)+(1/4) = 1/2

R(3)= P(X<=3) =P(X=1)+P(X=2) +P(X=3) = (1/4)+(1/4)+(1/4) = 3/4

R(4)= P(X<=4) =P(X=1)+P(X=2) +P(X=3) = 1

Y= No of heads observed after tossing the coin "x" times

={0,1,2,3,4}

P(Y=0)= P(X=1,Y=0)+ P(X=2,Y=0)+ P(X=3,Y=0)+ P(X=4,Y=0)

= (1/4)*(3/7) + (1/4)*(3/7) ² +(1/4)*(3/7) ³+ (1/4)*(3/7) ⁴

=0.1811745

P(Y=1)= P(X=1,Y=1)+ P(X=2,Y=1)+ P(X=3,Y=1)+ P(X=4,Y=1)

= (1/4)*(4/7) + (1/4)*(3/7)*(4/7) +(1/4)*(3/7) ² *(4/7) +(1/4)*(3/7) ³ *(4/7)

=

P(Y=2)= P(X=2,Y=2)+ P(X=3,Y=2)+ P(X=4,Y=2)

= (1/4)*(4/7)² + (1/4)*(3/7)*(4/7)² +(1/4)*(3/7) ² *(4/7)²  

=

P(Y=3)= P(X=3,Y=3)+ P(X=4,Y=3)

= (1/4)*(4/7)³ + (1/4)*(3/7)*(4/7)³

=

P(Y=4)= P(X=4,Y=4)

= (1/4)*(4/7)⁴ )

=

R(y)=P(Y<= y)

R(0) = P(y=0)

R(1)=P(X<=1) = P(y=0)+P(y=1)=

R(2)=P(X<=2)= P(y=0)+P(y=1)+P(y=2) =

R(3)= P(X<=3) =P(y=0)+P(y=1)+P(y=2)+P(y=3) =

R(4)= P(X<=4) =P(y=0)+P(y=1)+P(y=2)+P(y=3)+ P(y=4)= 1

b) X follows Uniform (0,4)

P(X=1) =1/4

P(X=2) =1/4

P(X=3) =1/4

P(X=4) =1/4

c) P(y | x=2)

P(y=0|x=2) = (3/7)²

P(y=1|x=2) = (4/7) * (3/7) * 2

P(y=2|x=2) = (4/7)²

=