Solution Found!
The conditional probability distribution of Y given X ??x
Chapter 5, Problem 18E(choose chapter or problem)
The conditional probability distribution of Y given X = x is \(f_{Y \mid x}(y)=x e^{-x y}\) for \(y>0\), and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10.
(a) Graph \(f_{Y \mid x}(y)=x e^{-x y}\) for \(y>0\) for several values of x. Determine:
(b) \(P(Y<2 \mid X=2)\)
(c) \(E(Y \mid X=2)\)
(d) \(E(Y \mid X=x)\)
(e) \(f_{X Y}(x, y)\)
(f) \(f_{Y}(y)\)
Questions & Answers
QUESTION:
The conditional probability distribution of Y given X = x is \(f_{Y \mid x}(y)=x e^{-x y}\) for \(y>0\), and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10.
(a) Graph \(f_{Y \mid x}(y)=x e^{-x y}\) for \(y>0\) for several values of x. Determine:
(b) \(P(Y<2 \mid X=2)\)
(c) \(E(Y \mid X=2)\)
(d) \(E(Y \mid X=x)\)
(e) \(f_{X Y}(x, y)\)
(f) \(f_{Y}(y)\)
ANSWER:Solution :
Step 1 of 6:
Given,
Then the density function of a uniform distribution is the reciprocal of the difference of the boundaries, on the interval between the boundaries or zero elsewhere.
=
We know that where b=10 and a=0.
The marginal probability distribution of X is
=
=
Our goal is:
a). We need to plot the graph .
b). We need to find P(Y<2/X=2).
c). We need to find E(Y/X=2).
d). We need to find E(Y/X=x).
e). We need to find .
a). We need to plot the graph .
We are using symbolab to plot the graph.
Then the graph is given below.
