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?
✓
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.
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.
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Find the volume of the parallelepiped determined by the vectors $\bar{a}, \bar{b}$ and $\bar{c}$ : $\quad \bar{a}=\hat{i}+2 \hat{j}+3 \hat{k}, \bar{b}=-\hat{i}+\hat{j}+2 \hat{k}, \bar{c}=2 \hat{i}+\hat{j}+4 \hat{k}$
→$\int \frac{1}{x \sin ^2(\log x)} d x$
→Evaluate:
→$\int \sqrt{1+\sin 2 x} \cdot d x$
Differentiate the following w. r. t. x.$\log \left(x^5+4\right)$
→Write the negations of the following :Polygon ABCDE is a pentagon.
Write the duals of each of the following. (p ∨ q) ∧ (r ∨ s)
$\bar{a} \times(\bar{b} \cdot \bar{c})$
→Write the negations of the following.It is cold and raining.
Verify that the combined equation of lines 2x + 3y = 0 and x – 2y = 0 is a homogeneous equation
of degree two.
of degree two.