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&nbsp;f(x,y)=ye−y(1+x),x,y&gt;0&nbsp;Give the marginal density functions for both random variables. Are the independent?

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A missile system&#039;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&amp;nbsp;f(x,y)=ye−y(1+x),x,y&amp;gt;0&amp;nbsp;Give the marginal density functions for both random variables. Are the independent?

odgovoreh · Accepted Answer

Step 1 f(x,y)=ye−y(1+x),x,y&amp;gt;0 Step 2 marginal density of X random variable: fx(x)=∫0∞ye−y(1+x)dy =1!(1+x)2   (∫0∞eaxxn−1dx=(n−1)!an) marginal density of Y random variable: fy(y)=∫0∞ye−y(1+x)dx =ye−y∫0∞e−yxdx =ye−y[−1ye−yx]0∞ =−e−y(0−1)=e−y Two random variables are independent iff f(x,y)=fx(x)⋅fy(y) =1(1+x)2⋅e−y≠f(x,y) Thus, X and Y are not independent.

Vasquez · Accepted Answer

Answer:The marginal density function of X is fX(x)=1+x, for x&amp;gt;0.The marginal density function of Y is fY(y)=1, for y&amp;gt;0.X and Y are notExplanation:Starting with X, we integrate the joint density function f(x, y) with respect to y from 0 to infinity:fX(x)=∫0∞f(x,y)dy=∫0∞ye−y(1+x)dyTo solve this integral, we can apply the property of the gamma function. Let&#039;s rewrite the integral in terms of the gamma function:fX(x)=∫0∞ye−y(1+x)dy=∫0∞(1+x)·(1+x)−1·y·e−y(1+x)dyWe can recognize that the term (1+x)−1 is the gamma function with parameter 1. Thus, the integral becomes:fX(x)=(1+x)∫0∞y1+x·e−y(1+x)dy=(1+x)·1=1+xTherefore, the marginal density function of X is:fX(x)=1+x,x&amp;gt;0Moving on to Y, we integrate the joint density function f(x, y) with respect to x from 0 to infinity:fY(y)=∫0∞f(x,y)dx=∫0∞ye−y(1+x)dxLet&#039;s simplify this integral by factoring out the common term y:fY(y)=y∫0∞e−y(1+x)dxTo solve this integral, we can use the exponential function&#039;s integral property:∫eaxdx=1aeaxApplying this property to the integral, we have:fY(y)=y[−1ye−y(1+x)]0∞=y(0−(−1y))=1Therefore, the marginal density function of Y is:fY(y)=1,y&amp;gt;0To 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:ye−y(1+x)=(1+x)·1Simplifying this equation:ye−y(1+x)=1+xThis equality does not hold for all values of x and y. Therefore, X and Y are not independent.

RizerMix · Accepted Answer

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)=∫0∞f(x,y)dy=∫0∞ye−y(1+x)dyTo evaluate this integral, we can use integration by parts. Let&#039;s set u=y and dv=e−y(1+x)dy. Then, du=dy and v=−11+xe−y(1+x). Applying the integration by parts formula:fX(x)=[−y1+xe−y(1+x)]0∞+∫0∞11+xe−y(1+x)dyThe first term evaluates to 0 at both bounds, so we only need to evaluate the integral:fX(x)=11+x∫0∞e−y(1+x)dyThis integral is the Laplace transform of the function e−y(1+x), which is equal to 11+x. Therefore, we have:fX(x)=11+xSimilarly, 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)=∫0∞f(x,y)dx=∫0∞ye−y(1+x)dxTo evaluate this integral, we can first factor out the constant y:fY(y)=y∫0∞e−y(1+x)dxNow, we can integrate with respect to x:fY(y)=−y[1ye−y(1+x)]0∞=[e−y(1+x)]0∞=0−(−e−y)=e−ySo, we have:fY(y)=e−yTo 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)=ye−y(1+x)≠fX(x)·fY(y)=11+x·e−ySince the joint density function is not equal to the product of the marginal density functions, X and Y are not independent.

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

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