Verify if the following functions are p.d.f. of a continuous r.v.… - Vidyadip

Question

Verify if the following functions are p.d.f. of a continuous r.v. $X$. : $f(x)=e^{-x}$, for $0<x<\infty$ and $=0$, otherwise.

✓

Answer

$e^{-x}$ is $\geq 0$ for any value of $x$ since $e>0$,
$
\begin{aligned}
& \therefore \quad e^{-x}>0, \text { for } 0<x<\infty \\
& \int_0^{\infty} f(x) d x=\int_0^{\infty} e^{-x} d x=\left[-e^{-x}\right]_0^{\infty}=\left[\frac{1}{e^{\infty}}-e^0\right]=-(0-1)=1
\end{aligned}
$
Both the conditions of p.d.f. are satisfied $f(x)$ is p.d.f. of r.v.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Answer the following questions in short." from Probability Distributions→Probability Distributions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

Write the negations of the following :Polygon ABCDE is a pentagon.

Find $k$, if the sum of the slopes of the lines represented by $x^2 + kxy − 3y^2 = 0$ is twice their product.

Find $\bar{a} \cdot \bar{b}$ if $|\bar{a}|=3,|\bar{b}|=\sqrt{6}$, the angle between $\bar{a}$ and $\bar{b}$ is $45^{\circ}$.

Compute the following sums:
$\begin{bmatrix}3&-2\\1&4\end{bmatrix}+\begin{bmatrix}-2&4\\1&3\end{bmatrix}$

Rewrite the following statements without using if .. then.It 2 is a rational number then $\sqrt{2}$ is irrational number.

$\int \frac{\cos 2 x}{\sin ^2 x} d x$

Obtain the differential euqation by eliminating the arbitrary constants from the following :
$y=A e^{3 x}+B e^{-3 x}$

Find mean for the following probability distribution.

X	0	1	2	3
P(X=x)	$\frac{1}{6}$	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{1}{6}$

Evaluate the following integrals:
$\int\log_\text{x}\text{xdx}$

Find the vector equation of the line passing through the point having position vector $4 \hat{i}-\hat{j}+2 \hat{k}$ and parallel to vector $-2 \hat{i}-\hat{j}+\hat{k}$