Problem 1: Computing a PDF from a CDF (25 points) (a) Find…
Problem 1: Computing a PDF from a CDF (25 points) (a) Find the PDF of the following CDF, where \(\phi(x)\) is the CDF of a standard Gaussian. \begin{align} F_X(x) = \begin{cases} 0 & x < 0 \\ \phi(x) & x \geq 0 \end{cases} \end{align} (b) Write out the definition of a median, and find the median of the random variable X. Problem 2: The Sum of Two Independent Random Variables (25 points) Let \(X \sim \text{Uniform}(-1,1)\) and \(Y\) have the PDF \(f_Y(y) = e^y\) for \(y < 0\). Let \(Z = X + Y\) and find the PDF of Z. Problem 3: Conditional Expectation (25 points) Let \(X \sim \text{Normal}(0,1)\) and \(Y | X=x \sim \text{Exponential}(|x|)\). Find \(\mathbb{E}\left[\frac{Y}{X} e^{Y|X|-Y}\right]\). Problem 4: The Transformation of Two Random Variables (25 points) Say \(X\) has the PDF \(f_X(x) = -2x\) for \(-1 < x < 0\) and let \(Y \sim \text{Uniform}(0,1)\). Set \(Z = X/Y\) and find the PDF of \(Z\). Congratulations, you are almost done with Midterm 2. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope to submit your work: Midterm 2 - US Students Return to this window and click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.
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