Problem 6 (ASV 3.37). Suppose that a random variable X has the cumulative distribution function x >0 1+x F(x) = x < 0. (a) Find the probability density function f. (b) Calculate P(2< X < 3). (c) Calculate E[(1+ X)²e-2X].
Problem 6 (ASV 3.37). Suppose that a random variable X has the cumulative distribution function x >0 1+x F(x) = x < 0. (a) Find the probability density function f. (b) Calculate P(2< X < 3). (c) Calculate E[(1+ X)²e-2X].
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![**Problem 6 (ASV 3.37):** Suppose that a random variable \( X \) has the cumulative distribution function
\[
F(x) =
\begin{cases}
\frac{x}{1+x}, & x \geq 0 \\
0, & x < 0
\end{cases}
\]
(a) Find the probability density function \( f \).
(b) Calculate \( P(2 < X < 3) \).
(c) Calculate \( E[(1 + X)^2 e^{-2X}] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F25ecc335-25be-4ebd-8f8c-629ee64b2629%2Fb4ef11a5-9067-4530-907f-438bd28dac4a%2Fmtx83k_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 6 (ASV 3.37):** Suppose that a random variable \( X \) has the cumulative distribution function
\[
F(x) =
\begin{cases}
\frac{x}{1+x}, & x \geq 0 \\
0, & x < 0
\end{cases}
\]
(a) Find the probability density function \( f \).
(b) Calculate \( P(2 < X < 3) \).
(c) Calculate \( E[(1 + X)^2 e^{-2X}] \).
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