Question

Let $Y$ be a continuous random variable whose PDF is as $g\left(y\right)=\frac{y^4}{125}$ for $yϵ(0,a)$. Constant "$a$" makes $g\left(y\right)$ a valid PDF. Find the value of $a^5$.

• A. $225$
• B. $625$
• C. $525$
• D. None of these

Answer 1

The correct option is B $625$

Let $Y$ be a continuous random variable whose PDF is as $g\left(y\right)=\frac{y^4}{125}$ for $y\in (0,a)$.

Also given that constant $a$ makes $g\left(y\right)$ a valid PDF.

We have to find the value of $a^5$.

We know that, a PDF is valid if it satisfies the following

$\left(1\right)$ $f\left(x\right)$ is positive everywhere.

$\left(2\right)$ The area under the curve $f\left(x\right)$ in the range $(a,b)$is $1$, that is: ${\int }_a^{b}f\left(x\right)dx=1$.

Thus we have ${\int }_0^{a}g\left(y\right)dy=1$

$\Rightarrow {\int }_0^a\frac{y^4}{125}dy=1$

$\Rightarrow \frac{1}{125}{\int }_0^a{y}^{4}dy=1$

$\Rightarrow \frac{1}{125}{\left[\frac{y^5}{5}\right]}_0^a=1$

$\Rightarrow \frac{1}{625}{\left[y^5\right]}_0^a=1$

$\Rightarrow \frac{1}{625}[a^5-0]=1$

$\Rightarrow \frac{a^5}{625}=1$

$\Rightarrow a^5=625$