Problem 3. Let X be an exponential random variable with rate \lambda. Let c be a positive constant. (a) Use the definition of conditional expectation to calculate $E(X|X > c)$. (b) Use the memoryless property to determine $E(X|X > c)$. (c) Use the definition of conditional expectation to calculate $E(X|X < c)$. (d) Prove the following identity $E(X) = E(X|X < c)P(X < c) + E(X|X > c)P(X > c)$. (e) Use the conclusion in (a) or (b) and the identity in (d) to calculate $E(X|X < c)$.
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First, we need to find the conditional probability density function (pdf) of X given X > c. The pdf of an exponential random variable X with rate A is given by: f_X(x) = A e^{-Ax} for x > 0 Now, we need to find the conditional pdf f_{X|X>c}(x). Using the Show more…
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