"How do I find the probability density function..." was answered by students like you

veksetz

Answered question

2022-01-16

How do I find the probability density function of a random variable X?

Answer & Explanation

Neunassauk8

Beginner2022-01-16Added 30 answers

If $X \sim F_{X} (x)$ , where $F_{X} (x)$ is the probability distribution function, then
$F_{x}^{'} (x) = f_{X} (x)$ where $f_{X} (x)$ is the probability density function.
Explanation:
By definition
$P (X \leq x) = F_{X} (x)$ where $F_{X} (x)$ is the distribution function of the random variable X.
This is sort of analogous to various areas of science where one might consider density as mass divided by volume $ρ = \frac{m}{v} .$
In physics if one were attempting to find how mass is distributed in an object for something like center of mass they would integrate it $x = \int_{Ω} ρ d A .$
Therein lies the analogy. Just like a physical object is a collection of particles, a probability space is a collection of outcomes.
So, if the probability distribution is described by $F_{X} (x)$ , then it would make sense that $F_{X} (x) = \int_{Ω} f_{X} (x) d x$ , where $f_{X} (x)$ is the probability density function.
So,
$F_{X} (x) = \int_{Ω} f_{X} (x) d x$
$\Leftrightarrow$
$F_{X}^{'} (x) = (\int_{Ω} f_{X} 9 x) d x)^{'} = f_{X} (x)$
$\Leftrightarrow$
$F_{X}^{'} (x) = f_{X} (x)$

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