A continuous random variable X has cumulative distribution function (CDF) F given by, \begin{equation*} F(x) = \begin{cases} 0, & x < 0 \\ x^3 & 0 \le x \le 1 \\ 1 & x > 1. \end{cases} \end{equation*} (a) Calculate the expectation E(X) of X. (b) Calculate the variance Var(X) of X.
Added by Johnny C.
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For 0 < x < 1, F(x) = x^3. So, f(x) = d(x^3)/dx = 3x^2. Now, we can calculate the expectation E(X) using the PDF: E(X) = ∫[x * f(x)]dx from 0 to 1 E(X) = ∫[x * 3x^2]dx from 0 to 1 E(X) = ∫[3x^3]dx from 0 to 1 E(X) = [0.75x^4] from 0 to 1 E(X) = 0.75(1)^4 - Show more…
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