Let X be a random variable with probability density function f(x) = begincases c1/sqrtx & if 0 < x le 1 c2/x^2 & if 1 < x < infty 0 & otherwise, endcases where c1 and c2 are appropriate real constants. If P(X in [frac14, 4]) = frac58, then consider the following statements: (I) P(X in [3,5]) = 1/12 (II) Both X as well as frac1X do not have finite expectations. Which of the above statements is/are true?

Q39Section ST (Statistics (Q11–Q65))39 of 65

Piecewise power density moments

Let $X$ be a random variable with probability density function $f(x) = \begin{cases} \frac{c_1}{\sqrt{x}} & \text{if } 0 < x \le 1 \ \frac{c_2}{x^2} & \text{if } 1 < x < \infty \ 0 & \text{otherwise,} \end{cases}$ where $c_1$ and $c_2$ are appropriate real constants. If $P\left(X \in \left[\frac14, 4\right]\right) = \frac58$ , then consider the following statements: (I) $P(X \in [3,5]) = \frac{1}{12}$ (II) Both $X$ as well as $\frac1X$ do not have finite expectations. Which of the above statements is/are true?

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