Question-37

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Let the joint probability density function of \(X\) and \(Y\) be given as:

\(f(x, y) = \begin{cases} e^{-(x + y)}, & x \geq 0, y \geq 0 \\ 0, & \text{otherwise} \end{cases}\)

What is \(P(X + Y \leq 1)?\)

\(1-e^{-2}\)
\(1-2e^{-1}\)
\(e^{-2}\)
\(1-e^{-4}\)

Hint

The inequality \(X+Y \leq1\) represents a boundary in the \(xy\)-plane. In this case, it corresponds to a triangular region where the sum of \(X\) and \(Y\) is less than or equal to 1.

Find the limits of \(x\) and \(y\) and do the integration to obtain the required probability.

Answer

\(1-e^{-2}\)
\(1-2e^{-1}\)
\(e^{-2}\)
\(1-e^{-4}\)

Solution