Name a continuous and discrete probability distribution, write the CDF, PDF, MGF, mean, variance, median, and mode of that, then provide a real-life example.
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Key Concepts
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Examples A continuous random variable X has the CDF given by F(x) = 0 if x < 1 2(x - 2)^4 if 1 < x < 2 1 if x > 2 Find the pdf of X. Find E(X). Find V(X).
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Let $X$ have the $\operatorname{cdf} F(x)$ that is a mixture of the continuous and discrete types, namely $$ F(x)=\left\{\begin{array}{ll} 0 & x<0 \\ \frac{x+1}{4} & 0 \leq x<1 \\ 1 & 1 \leq x \end{array}\right. $$ Determine reasonable definitions of $\mu=E(X)$ and $\sigma^{2}=\operatorname{var}(X)$ and compute each. Hint: Determine the parts of the pmf and the pdf associated with each of the discrete and continuous parts, and then sum for the discrete part and integrate for the continuous part.
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