Given that x is a random variable in range -0- - - with a probability density function e-x-2-K- the value of the constant K is -

For p.d.f, ∫_ ∞^∞f(x)dx=1 ⇒ ∫_0^∞e^ x2K dx=1 ⇒ 1K ( e^ x2 ( 12 ) )_0^∞=1 ⇒ 2K ( e^ x2 )_0^∞=1 ⇒ 2K(0 1)=1 ⇒ K=2 nbsp; ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Other

Engineering Mathematics

Discrete and Continuous RV

Given that x ...

Question

Given that x is a random variable in range $[0, \infty]$ with a probability density function $e^{- x / 2} / K,$ the value of the constant K is ______ .

2

Open in App

Solution

The correct option is A 2
For p.d.f, $\int_{- \infty}^{\infty} f (x) d x = 1$

$\Rightarrow \int_{0}^{\infty} \frac{e^{\frac{- x}{2}}}{K} d x = 1$

$\Rightarrow \frac{1}{K} {⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{e^{\frac{- x}{2}}}{(- \frac{1}{2})} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}_{0}^{\infty} = 1$

$\Rightarrow - \frac{2}{K} {⎛ ⎝ e^{- \frac{x}{2}} ⎞ ⎠}_{0}^{- \frac{x}{2}} = 1$

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

Suggest Corrections

Similar questions

Given that x is a random variable in range $[0, \infty]$ with a probability density function $e^{- x / 2} / K,$ the value of the constant K is ______ .

Q. If the probability density function of a random variable is given by,

f (x) = {\begin{matrix} k (1 - x^{2}), & 0 < x < 1 0, & elsewhere \end{matrix}

find

k

and the distribution function of the random variable.

Q. If X is a continuous random variable whose probability density function is given by

f (x) = (\begin{matrix} k (5 x - 2 x^{2}), & 0 \leq x \leq 2 0, & otherwise \end{matrix}

Then P(x > 1) is

Q. A random variable X has a probability density function

f (x) = (\begin{matrix} {ke}^{n} e^{- x}; & x \geq 0 0; & otherwise \end{matrix}

(n is an integer)
with mean 3. The values of {k, n} are

Q. The probability density function (PDF) of a random variable, X is given by,

f_{X} (x) = K e^{- \frac{(x + 1)^{2}}{18}}

The value of K will be______.

Join BYJU'S Learning Program

Given that x is a random variable in range [0, ∞] with a probability density function e−x/2/K, the value of the constant K is ______ . 2

The correct option is A 2For p.d.f, ∫∞−∞f(x)dx=1 ⇒ ∫∞0e−x2K dx=1 ⇒ 1K⎛⎜ ⎜ ⎜ ⎜⎝e−x2(−12)⎞⎟ ⎟ ⎟ ⎟⎠∞0=1 ⇒ −2K⎛⎝e−x2⎞⎠∞0=1 ⇒ −2K(0−1)=1 ⇒ K=2

Given that x is a random variable in range $[0, \infty]$ with a probability density function $e^{- x / 2} / K,$ the value of the constant K is ______ .

2