Search any question & find its solution
Question:
Answered & Verified by Expert
If a continuous random variable $\mathrm{X}$ has probability density function $\mathrm{f}(x)$ given by
$$
f(x)=\left\{\begin{array}{cc}
a x & , \text { if } 0 \leq x < 1 \\
a & , \text { if } 1 \leq x < 2 \\
3 a-a x & \text { if } 2 \leq x \leq 3 \\
0 & \text {, otherwise }
\end{array},\right.
$$
then a has the value
Options:
$$
f(x)=\left\{\begin{array}{cc}
a x & , \text { if } 0 \leq x < 1 \\
a & , \text { if } 1 \leq x < 2 \\
3 a-a x & \text { if } 2 \leq x \leq 3 \\
0 & \text {, otherwise }
\end{array},\right.
$$
then a has the value
Solution:
1224 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{2}$
Since $\mathrm{f}(x)$ is the p.d.f. of $\mathrm{X}$,
$$
\begin{aligned}
& \int_{-\infty}^{\infty} \mathrm{f}(x) \mathrm{d} x=1 \\
& \Rightarrow \int_0^1 \mathrm{a} x \mathrm{~d} x+\int_1^2 \mathrm{ad} x+\int_2^3(3 \mathrm{a}-\mathrm{ax}) \mathrm{d} x=1 \\
& \Rightarrow \mathrm{a}\left[\frac{x^2}{2}\right]_0^1+\mathrm{a}[x]_1^2+\left[3 \mathrm{a} x-\frac{\mathrm{a} x^2}{2}\right]_2^3=1
\end{aligned}
$$
$\begin{aligned} & \Rightarrow \mathrm{a}\left(\frac{1}{2}\right)+\mathrm{a}(1)+\left(\frac{9 \mathrm{a}}{2}-4 \mathrm{a}\right)=1 \\ & \Rightarrow 2 \mathrm{a}=1 \\ & \Rightarrow \mathrm{a}=\frac{1}{2}\end{aligned}$
$$
\begin{aligned}
& \int_{-\infty}^{\infty} \mathrm{f}(x) \mathrm{d} x=1 \\
& \Rightarrow \int_0^1 \mathrm{a} x \mathrm{~d} x+\int_1^2 \mathrm{ad} x+\int_2^3(3 \mathrm{a}-\mathrm{ax}) \mathrm{d} x=1 \\
& \Rightarrow \mathrm{a}\left[\frac{x^2}{2}\right]_0^1+\mathrm{a}[x]_1^2+\left[3 \mathrm{a} x-\frac{\mathrm{a} x^2}{2}\right]_2^3=1
\end{aligned}
$$
$\begin{aligned} & \Rightarrow \mathrm{a}\left(\frac{1}{2}\right)+\mathrm{a}(1)+\left(\frac{9 \mathrm{a}}{2}-4 \mathrm{a}\right)=1 \\ & \Rightarrow 2 \mathrm{a}=1 \\ & \Rightarrow \mathrm{a}=\frac{1}{2}\end{aligned}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.