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Probability Distributions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsProbability Distributions2 Marks
MCQ
The p.d.f. of a continuous random variable X is
$\mathrm{f}(x)=\frac{\mathrm{K}}{\sqrt{x}} ; \quad 0<x<4 \\ =0$; otherwise
Then $\mathrm{P}(\mathrm{X} \geq 1)$ is equal to
  • A
    0.2
  • B
    0.3
  • C
    0.4
  • ✓
    0.5
✓

Answer

Correct option: D.
0.5
(D)
$\int_{-\infty}^{\infty} \mathrm{f}(x) \mathrm{d} x=1$
$\therefore \quad \int_0^4 \frac{\mathrm{~K}}{\sqrt{x}} \mathrm{~d} x=1$
$\begin{aligned} & \Rightarrow \mathrm{K}[2 \sqrt{x}]_0^4=1 \\ & \Rightarrow 2 \mathrm{~K}[\sqrt{4}-\sqrt{0}]=1 \\ & \Rightarrow 4 \mathrm{~K}=1 \\ & \Rightarrow \mathrm{~K}=\frac{1}{4}\end{aligned}$
$\therefore \quad P(X \geq 1)=P(1 \leq X<4)$
$\begin{aligned} & =\int_1^4 \mathrm{f}(x) \mathrm{d} x=2 \mathrm{~K}[\sqrt{x}]_1^4 \\ & =2 \times \frac{1}{4}(2-1) \\ & =\frac{1}{2}=0.5\end{aligned}$

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