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Probability Distributions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsProbability Distributions1 Mark
Question
Let $X$ be a continuous random variable whose probability density function is $f(x)=\frac{x^3}{4}$ for an interval $0<x<c$. What is the value of the constant $\mathrm{c}$ that makes $f(x)$ a valid probability density function?
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Answer

Note that the integral of the p. d. f. over the support of the random variable must be That is, $\int_0^c f(x) d x=1$.
That is, $\int_0^c\left(\frac{x^3}{4}\right) d x=1$. But, $\int_0^c\left(\frac{x^3}{4}\right) d x=\left[\frac{x^4}{16}\right]_0^c=\frac{c^4}{16}$. Since this integral must be equal to 1 , we have $\frac{c^4}{16}=1$, or equivalently $c^4=16$, so that $c=2$ since $\mathrm{c}$ must be positive.

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