Tools

Discrete and Continuous Random Variables

1. Let XX be a contiuous RV with PDF fX(x)={x2−120≤x≤10otherwisef_X(x) = \begin{cases} x^2-\frac{1}{2} & 0 \leq x \leq 1 \\ 0 & otherwise \end{cases}. For n∈Nn\in N, calculate E[Xn]E[X^n].
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2. Let XX be a contiuous RV with PDF fX(x)={x2−120≤x≤10otherwisef_X(x) = \begin{cases} x^2-\frac{1}{2} & 0 \leq x \leq 1 \\ 0 & otherwise \end{cases}. Find Var(X)Var(X)

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3. Let fX(x)={ce−3xx≥00otherwisef_X(x) = \begin{cases} ce^{-3x} & x \geq 0 \\ 0 & otherwise \end{cases}. Find value of cc such that fX(x)f_X(x) is a valid PDF
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4. Find P(1<X<3)P(1<X<3) for the above RV in question 3

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5. Let XX be a uniform discrete RV with range RX={0,π4,π2,3π4,π}\mathbb{R}_X = \{0,\frac{\pi}{4},\frac{\pi}{2},\frac{3\pi}{4},\pi\}. Find E[sin(X)]E[sin(X)]
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