Probability Density Function
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In this section, we will understand Probability Density Function of a Continuous Random Variable.
Probability Density Function(PDF):
This is a function used for continuous random variables to describe the likelihood of the variable taking on a value
within a specific range or interval.
Since, at any given point the probability of a continuous random variable is zero,
we find the probability within a given range.
Note: Called ‘density’ because probability is spread continuously over a range of values
rather than being concentrated at a single point as in PMF.
e.g: Uniform, Gaussian, Exponential, etc.
Note: PDF is a continuous function \(f(x)\).
It is also the derivative of Cumulative Distribution Function (CDF) \(F_X(x)\)
\(PDF = f(x) = F'(X) = \frac{dF_X(x)}{dx} \)
Key properties of PDF
- Non-Negative: Function must be non-negative everywhere i.e \(f(x) \ge 0 \forall x\).
- Sum = 1: Total area under curve must be equal to 1.
\( \int_{-\infty}^{\infty} f(x) \,dx = 1\) - Probability of a continuous random variable in the range [a,b] is given by -
\( P(a \le x \le b) = \int_{a}^{b} f(x) \,dx\)
Note
We use a general term Probability Distribution Function for both PMF(discrete) and PDF(continuous) because both describe how the probability is distributed across a random variable’s entire domain.For example:
Consider a line segment/interval from \(\Omega = [0,2] \)
Random variable \(X(\omega) = \omega\)
i.e \(X(1) = 1 ~and~ X(1.1) = 1.1 \)
Note: If we know the PDF of a continuous random variable, then we can find the probability of any given region/interval
by calculating the area under the curve.
End of Section