1. Meaning of a Probability Density Function
A Probability Density Function (PDF) describes how probability is spread over the values of a continuous random variable. Instead of assigning probability to individual values, a PDF gives a curve whose area represents probability.
For a continuous variable X, the PDF is written as \( f(x) \).
The probability that X lies between two numbers a and b is found using the area under the curve:
\( P(a \le X \le b) = \int_a^b f(x)\,dx \)
This means a PDF gives a density, not a direct probability at a single point.
2. Why Individual Values Have Zero Probability
For a continuous random variable, the probability that X takes any exact value (like X = 5.2) is always zero:
\( P(X = x) = 0 \)
Instead, probability only makes sense over a range, such as \( P(5 < X < 6) \).
3. Properties of a PDF
A PDF must satisfy the following conditions:
- Non-negativity:
\( f(x) \ge 0 \)
- Total area under the curve is 1:
\( \int_{-\infty}^{\infty} f(x)\,dx = 1 \)
- Probability of an interval comes from area:
\( P(a < X < b) = \int_a^b f(x)\,dx \)
4. Example 1: Uniform Distribution on an Interval
Let X be a continuous variable uniformly distributed over the interval [0, 2]. The PDF is:
\( f(x) = 1/2 \quad \text{for } 0 \le x \le 2 \)
It is 0 outside this interval.
The total area is:
\( \int_0^2 1/2\,dx = 1 \)
Probability that X lies between 0.5 and 1.5:
\( P(0.5 < X < 1.5) = \int_{0.5}^{1.5} 1/2\,dx = 1/2 \)
5. Example 2: A Simple Piecewise PDF
Consider a PDF defined by:
\( f(x) = x \quad \text{for } 0 \le x \le 1 \)
It is 0 outside this range.
Total area:
\( \int_0^1 x\,dx = 1/2 \)
This is not 1, so the function must be scaled:
\( f(x) = 2x \quad \text{for } 0 \le x \le 1 \)
Now the total area is:
\( \int_0^1 2x\,dx = 1 \)
Finding the probability that X lies between 0.3 and 0.8:
\( P(0.3 < X < 0.8) = \int_{0.3}^{0.8} 2x\,dx = (0.64 - 0.09) = 0.55 \)