Question

A continuous random variable X has the P.D.F $f\left( x \right)=3{{x}^{2}},0\le x\le 1$. The value of a constant $\lambda $ that satisfies the relation $P\left\{ X\le \lambda \right\}=P\left\{ X>\lambda \right\}$ is
A. ${{\left( \dfrac{1}{3} \right)}^{\dfrac{1}{2}}}$
B. ${{\left( \dfrac{1}{2} \right)}^{\dfrac{1}{3}}}$
C. ${{\left( \dfrac{2}{3} \right)}^{\dfrac{1}{2}}}$
D. ${{\left( \dfrac{2}{3} \right)}^{\dfrac{1}{3}}}$

Answer 1

Hint: We first check if the given P.D.F satisfies the condition of \[\int\limits_{-\infty }^{\infty }{f\left( x \right)dx}=1\]. We take the integration of \[\int\limits_{0}^{1}{3{{x}^{2}}dx}\]. We then use the given condition of $P\left\{ X\le \lambda \right\}=P\left\{ X>\lambda \right\}$ to form the integration part. We solve the equation to find the final solution.

Complete step by step answer:
The given P.D.F is $f\left( x \right)=3{{x}^{2}},0\le x\le 1$. The condition for P.D.F is \[\int\limits_{-\infty }^{\infty }{f\left( x \right)dx}=1\].
Here we take the integration for $0\le x\le 1$.
We get \[\int\limits_{0}^{1}{3{{x}^{2}}dx}=1\].
\[\begin{align}
  & \int\limits_{0}^{1}{3{{x}^{2}}dx}=1 \\
 & \Rightarrow \left[ {{x}^{3}} \right]_{0}^{1}=1-0=1 \\
\end{align}\]
The given equation satisfies the condition.
It is given that $P\left\{ X\le \lambda \right\}=P\left\{ X>\lambda \right\}$. We express them in the form of integration and get
\[\begin{align}
  & P\left\{ X\le \lambda \right\}=P\left\{ X>\lambda \right\} \\
 & \Rightarrow \int\limits_{0}^{\lambda }{3{{x}^{2}}dx}=\int\limits_{\lambda }^{1}{3{{x}^{2}}dx} \\
 & \Rightarrow \left[ {{x}^{3}} \right]_{0}^{\lambda }=\left[ {{x}^{3}} \right]_{\lambda }^{1} \\
\end{align}\]
We now solve the equation and get
\[\begin{align}
  & \left[ {{x}^{3}} \right]_{0}^{\lambda }=\left[ {{x}^{3}} \right]_{\lambda }^{1} \\
 & \Rightarrow {{\lambda }^{3}}-0=1-{{\lambda }^{3}} \\
 & \Rightarrow 2{{\lambda }^{3}}=1 \\
 & \Rightarrow {{\lambda }^{3}}=\dfrac{1}{2} \\
 & \Rightarrow \lambda ={{\left( \dfrac{1}{2} \right)}^{\dfrac{1}{3}}} \\
\end{align}\]
Therefore, the correct option is B.

Note: It is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. For cases of P.M.F we have to use the distinct values of the masses.