Let X be a random variable with probability density function f(x) = begincases Gamma(alpha+beta)/Gamma(alpha)Gamma(beta) x^alpha-1(1-x)^beta-1 & if 0 < x < 1 0 & otherwise, endcases where alpha 0, beta 0. If E(X) = 1/3 and E(X^2) = 1/6, then alpha + 3beta equals

Q19Section ST (Statistics (Q11–Q65))19 of 65

Beta moments pin down parameters

Let $X$ be a random variable with probability density function $f(x) = \begin{cases} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1}(1-x)^{\beta-1} & \text{if } 0 < x < 1 \ 0 & \text{otherwise,} \end{cases}$ where $\alpha > 0, \beta > 0$ . If $E(X) = \frac{1}{3}$ and $E(X^2) = \frac{1}{6}$ , then $\alpha + 3\beta$ equals

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