Suppose that X and Y are continuous random variables with joint pdf f(x,y)=e−(x+y)0&lt;x&lt;∞ and 0&lt;y&lt;∞ and zero otherwise. Find P(X+Y&gt;3)

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Suppose that X and Y are continuous random variables with joint pdf f(x,y)=e−(x+y)0&amp;lt;x&amp;lt;∞ and 0&amp;lt;y&amp;lt;∞ and zero otherwise.  Find P(X+Y&amp;gt;3)

Nola Robson · Accepted Answer

Step 1  Introduction:  The joint density function of two random variables X and Y is given below:  f(x,y)={beg∈{matrix}e−(x+y);   0≤x≤∞,0≤y&amp;lt;∞∖0; elsewhereend{matrix}}  The marginal density function of X is,  f(x)=∫0∞e−(x+y)dy  =e−x∫0∞e−ydy  =−e−x[−e−y]0∞  =−e−x[e−∞−e−0]  =−e−x[0,1]  =e−x  Step 2  The probability of P(X + Y &amp;gt; 3) is obtained as 0.1991 from the calculation given below:  P(X+Y&amp;gt;3)=1−P(X+Y≤3)  =1−∫03∫03−xe−(x+y)dxdy  =1−∫03∫03−xe−xe−ydxdy  =1−∫03e−xdx∫03−xe−ydy  =1−∫03e−xdx[−e−y]03x  =1−(−∫03e−xdx[e−(3−x)−e−0])  =1−(−∫03e−xdx[e−3+x−1])

Suppose that X and Y are continuous random variables with joint pdf f(x,y)=e^

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