The p. d. f. of a random variable x is given by fx=1/4a,0 0 =0, otherwise and Px 5a/2 then K = …….

Correct answer: (A) 1. We have p. d. f. of a random variable x is given by fx=1/4a,0 0 and also, px 5a/2 ⇒∫03a/2fxdx=k∫5a/24afxdx ⇒∫03a/21/4adx=k∫5a/24a1/4adx ⇒1/4a∫03a/2dx=k/4a∫5a/24adx ⇒x03a/2=kx5a/24a ⇒3a/2-0=k4a-5a/2 ⇒3a/2=k3a/2 ⇒k=1

Q54 / 150 · MHT CET 2019 (Shift 2) · Mathematics

The p. d. f. of a random variable x is given by $f (x) = \frac{1}{4 a}, 0 < x < 4 a, (a > 0)$ =0, otherwise and $P (x < \frac{3 a}{2}) = k P (x > \frac{5 a}{2})$ then K = …….

Correct answer
A
1
B
$\frac{1}{2}$
C
$\frac{1}{8}$
D
$\frac{3}{2}$

Solution

We have p. d. f. of a random variable x is given by

f (x) = \frac{1}{4 a}, 0 < x < 4 a . (a > 0)

and also,

p (x < \frac{3 a}{2}) = k p (x > \frac{5 a}{2})

\Rightarrow \int_{0}^{\frac{3 a}{2}} f (x) d x = k \int_{\frac{5 a}{2}}^{4 a} f (x) d x

\Rightarrow \int_{0}^{\frac{3 a}{2}} (\frac{1}{4 a}) d x = k \int_{\frac{5 a}{2}}^{4 a} (\frac{1}{4 a}) d x

\Rightarrow \frac{1}{4 a} \int_{0}^{\frac{3 a}{2}} d x = \frac{k}{4 a} \int_{\frac{5 a}{2}}^{4 a} d x

\Rightarrow {(x)}_{0}^{\frac{3 a}{2}} = k {(x)}_{\frac{5 a}{2}}^{4 a}

\Rightarrow (\frac{3 a}{2} - 0) = k (4 a - \frac{5 a}{2})

\Rightarrow \frac{3 a}{2} = k (\frac{3 a}{2})

\Rightarrow k = 1

Hence, the correct answer is option: 'A'.

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Solution

Relevant Questions

The p. d. f. of a random variable x is given by $f (x) = \frac{1}{4 a}, 0 < x < 4 a, (a > 0)$ =0, otherwise and $P (x < \frac{3 a}{2}) = k P (x > \frac{5 a}{2})$ then K = …….