Name: Variance and Standard Deviation
Uploaded: 2019-07-19
Duration: 8 min 1 s
Description: What do you understand by standard deviation? In this video, we will learn how to calculate variance and standard deviation. Let us study this video and find...

Question

Find

k

if the function

f (x)

is defined by

\{\begin{cases} f (x) = k x (1 - x), & for 0 < x < 1 \\ = 0, & otherwise, \end{cases}

is the probability density function (p.d.f.) of a random variable (r.v.)

X

. Also find

P (x < \frac{1}{2})

.

Accepted Answer

Given that the function $\{\begin{cases} f (x) = k x (1 - x), & for 0 < x < 1 \\ = 0, & otherwise, \end{cases}$ is the p.d.f. of $X$ .

$\Rightarrow \int_{- \infty}^{\infty} f (x) d x = 1$

$\Rightarrow \int_{- \infty}^{0} f (x) d x + \int_{0}^{1} f (x) d x + \int_{1}^{\infty} f (x) d x = 1$

$\Rightarrow 0 + \int_{0}^{1} k x (1 - x) d x + 0 = 1$

$\Rightarrow k \int_{0}^{1} (x - x^{2}) d x = 1$

$\Rightarrow k {[\frac{x^{2}}{2} - \frac{x^{3}}{3}]}_{0}^{1} = 1$

$\Rightarrow k (\frac{1}{2} - \frac{1}{3} - 0) = 1$

$\Rightarrow \frac{k}{6} = 1$

Hence, $k = 6$

Now, we need to find: $P (x < \frac{1}{2})$

We know that

$P (x < \frac{1}{2}) = \int_{- \infty}^{\frac{1}{2}} f (x) d x$

$= \int_{- \infty}^{0} f (x) d x + \int_{0}^{\frac{1}{2}} f (x) d x$

$= 0 + \int_{0}^{\frac{1}{2}} k x (1 - x) d x$

$= k \int_{0}^{\frac{1}{2}} (x - x^{2}) d x$

$= k {[\frac{x^{2}}{2} - \frac{x^{3}}{3}]}_{0}^{\frac{1}{2}}$

$= k [\frac{1}{8} - \frac{1}{24} - 0]$

$= k (\frac{2}{24})$

$= 6 \times \frac{2}{24}$

Hence, $P (x < \frac{1}{2}) = \frac{1}{2}$ .

$x$	$0$	$1$	$2$	$3$	$4$
$= \int_{- \infty}^{0} f (x) d x + \int_{0}^{\frac{1}{2}} f (x) d x$	$0.45$	$0.35$	$0.15$	$0.03$	$0.02$

Find $k$ if the function $f (x)$ is defined by $\{\begin{cases} f (x) = k x (1 - x), & for 0 < x < 1 \\ = 0, & otherwise, \end{cases}$ is the probability density function (p.d.f.) of a random variable (r.v.) $X$ . Also find $P (x < \frac{1}{2})$ .

Important Questions on Probability Distribution

Learn and Prepare for any exam you want !

Find k if the function fx is defined by fx=kx1-x,for 0<x<1=0,otherwise,is the probability density function (p.d.f.) of a random variable (r.v.) X. Also find Px<12.

Important Questions on Probability Distribution

Learn and Prepare for any exam you want !

Find $k$ if the function $f (x)$ is defined by $\{\begin{cases} f (x) = k x (1 - x), & for 0 < x < 1 \\ = 0, & otherwise, \end{cases}$ is the probability density function (p.d.f.) of a random variable (r.v.) $X$ . Also find $P (x < \frac{1}{2})$ .