GATE (TF) Textile 2022 Question Paper Solution | GATE/2022/TF/26

Question 26 (Textile Technology & Fibre Science)

If the probability density function of a continuous random variable X is given by

$f(x)=\begin{cases}(x-2)a; & 2\leq x\leq4 \\(8-x)a, & 4\leq x\leq 8, \: where \: a \: is \: a \:constant \\0 & otherwise,\end{cases}$

then the value of $a$ is

(A)	$0.2$
(B)	$0.1$
(C)	$0.5$
(D)	$0.4$

[Show Answer]

Probability density function=1

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

$\Rightarrow \int_{-\infty }^{2} f(x) dx + \int_{2}^{4} f(x) dx + \int_{4}^{8} f(x) dx + \int_{8}^{\infty} f(x) dx = 1$

$\Rightarrow 0 + \int_{2}^{4} f(x) dx + \int_{4}^{8} f(x) dx + 0 =1$

$\Rightarrow \int_{2}^{4} f(x) dx + \int_{4}^{8} f(x) dx =1$

$\Rightarrow \int_{2}^{4} (x-2)adx + \int_{4}^{8} (8-x) adx =1$

$\Rightarrow a \left \{ \left [ \frac{x^2}{2} -2x \right ]_2^4 + \left [ 8x-\frac{x^2}{2} \right ]_4^8 \right \} = 1$

$\Rightarrow a \left \{ \left [ \left ( \frac{4^2}{2} - (2 \times 4) \right ) - \left ( \frac{2^2}{2}-(2 \times 2) \right )\right ] + \left [ \left ( (8 \times 8) - \frac{8^2}{2} \right ) - \left ( (8 \times 4 ) - \frac{4^2}{2} \right )\right ] \right \} = 1$

$\Rightarrow a \left \{ \left [ \left ( \frac{16}{2} - 8 \right ) - \left ( 2-4 \right )\right ] + \left [ \left ( 64 - \frac{64}{2} \right ) - \left ( 32 - \frac{16}{2} \right )\right ] \right \} = 1$

$\Rightarrow a \left \{ 0 + 2 + 32-24 \right \} = 1$

$\Rightarrow a \left \{ 10 \right \} =1$

$\Rightarrow a = 0.1$

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What is probability density function ?

A probability density function (pdf) is a mathematical function that describes the likelihood of obtaining the values of a continuous random variable within a certain range. It provides a continuous analogue to the discrete notion of a probability mass function, which is used for discrete random variables. The pdf assigns a probability to each value of the random variable, with the probabilities for all values summing to 1. The height of the pdf at a given value represents the probability density, or likelihood, of obtaining that value. The pdf is a fundamental tool in probability theory and statistics, and is used in a wide range of applications, including statistical modeling, estimation, and hypothesis testing.

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