Two electronic components of a missile system work in harmony for the success of the total system. Let X and Y denote the life in hours of the two com

Wribreeminsl

Wribreeminsl

Answered question

2021-05-05

A missile system's two electronic components must cooperate for the system as a whole to succeed. Let X and Y represent the combined life of the two components in hours. The joint density of X and Y is 
f(x,y)=yey(1+x),x,y>0 
Give the marginal density functions for both random variables. Are the independent?

Answer & Explanation

odgovoreh

odgovoreh

Skilled2021-05-06Added 107 answers

Step 1
f(x,y)=yey(1+x),x,y>0
Step 2
marginal density of X random variable:
fx(x)=0yey(1+x)dy
=1!(1+x)2   (0eaxxn1dx=(n1)!an)
marginal density of Y random variable:
fy(y)=0yey(1+x)dx
=yey0eyxdx
=yey[1yeyx]0
=ey(01)=ey
Two random variables are independent iff
f(x,y)=fx(x)fy(y)
=1(1+x)2eyf(x,y)
Thus, X and Y are not independent.

Vasquez

Vasquez

Expert2023-05-14Added 669 answers

Answer:
The marginal density function of X is fX(x)=1+x, for x>0.
The marginal density function of Y is fY(y)=1, for y>0.
X and Y are not
Explanation:
Starting with X, we integrate the joint density function f(x, y) with respect to y from 0 to infinity:
fX(x)=0f(x,y)dy=0yey(1+x)dy
To solve this integral, we can apply the property of the gamma function. Let's rewrite the integral in terms of the gamma function:
fX(x)=0yey(1+x)dy=0(1+x)·(1+x)1·y·ey(1+x)dy
We can recognize that the term (1+x)1 is the gamma function with parameter 1. Thus, the integral becomes:
fX(x)=(1+x)0y1+x·ey(1+x)dy=(1+x)·1=1+x
Therefore, the marginal density function of X is:
fX(x)=1+x,x>0
Moving on to Y, we integrate the joint density function f(x, y) with respect to x from 0 to infinity:
fY(y)=0f(x,y)dx=0yey(1+x)dx
Let's simplify this integral by factoring out the common term y:
fY(y)=y0ey(1+x)dx
To solve this integral, we can use the exponential function's integral property:
eaxdx=1aeax
Applying this property to the integral, we have:
fY(y)=y[1yey(1+x)]0=y(0(1y))=1
Therefore, the marginal density function of Y is:
fY(y)=1,y>0
To determine if X and Y are independent, we need to check if the joint density function factors into the product of the marginal density functions:
f(x,y)=fX(x)·fY(y)
Substituting the marginal density functions we found earlier:
yey(1+x)=(1+x)·1
Simplifying this equation:
yey(1+x)=1+x
This equality does not hold for all values of x and y. Therefore, X and Y are not independent.
RizerMix

RizerMix

Expert2023-05-14Added 656 answers

To find the marginal density functions for random variables X and Y, we need to integrate the joint density function f(x, y) over the appropriate ranges.
The marginal density function for X can be found by integrating f(x, y) with respect to y, while considering the range of valid values for y:
fX(x)=0f(x,y)dy=0yey(1+x)dy
To evaluate this integral, we can use integration by parts. Let's set u=y and dv=ey(1+x)dy. Then, du=dy and v=11+xey(1+x). Applying the integration by parts formula:
fX(x)=[y1+xey(1+x)]0+011+xey(1+x)dy
The first term evaluates to 0 at both bounds, so we only need to evaluate the integral:
fX(x)=11+x0ey(1+x)dy
This integral is the Laplace transform of the function ey(1+x), which is equal to 11+x. Therefore, we have:
fX(x)=11+x
Similarly, we can find the marginal density function for Y by integrating f(x, y) with respect to x, while considering the range of valid values for x:
fY(y)=0f(x,y)dx=0yey(1+x)dx
To evaluate this integral, we can first factor out the constant y:
fY(y)=y0ey(1+x)dx
Now, we can integrate with respect to x:
fY(y)=y[1yey(1+x)]0=[ey(1+x)]0=0(ey)=ey
So, we have:
fY(y)=ey
To determine if X and Y are independent, we need to check if their joint density function can be expressed as the product of their marginal density functions. In this case:
f(x,y)=yey(1+x)fX(x)·fY(y)=11+x·ey
Since the joint density function is not equal to the product of the marginal density functions, X and Y are not independent.

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