Let $Z \sim N(0,1)$. Define the clipped (truncated) variable $X = \min(\max(Z, -2),\, 2),$ so $X$ equals $Z$ if $|Z| \le 2$, equals $-2$ if $Z 2$. 1. Compute $E[X]$ and $\text{Var}(X)$ in closed form, expressing your answer in terms of the standard normal pdf $\phi$ and cdf $\Phi$. 2. Without a c…

Mean and Variance of a Clipped Normal

Expectation · Medium · Free problem

Let $Z \sim N(0,1)$. Define the clipped (truncated) variable $X = \min(\max(Z, -2),\, 2),$ so $X$ equals $Z$ if $|Z| \le 2$, equals $-2$ if $Z < -2$, and equals

$ if $Z > 2$. 1. Compute $E[X]$ and $\text{Var}(X)$ in closed form, expressing your answer in terms of the standard normal pdf $\phi$ and cdf $\Phi$. 2. Without a calculator, approximate $\text{Var}(X)$ to two decimal places, showing your whiteboard steps.

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