9231 · 4.1
Continuous random variables
A continuous random variable can take any value in a given range, like height or time. We use a function called a Probability Density Function (PDF) to describe the likelihood of the variable falling within a particular interval, where probability is found by calculating the area under the curve.
Need to know
What you need to know
- A CRV can take any value in a continuous interval.
- The probability density function, f(x), describes the distribution.
- Probability is the area under the PDF curve: P(a ≤ X ≤ b) = ∫[a,b] f(x) dx.
- The probability of a CRV equalling a specific value is always zero: P(X = c) = 0.
Explanation
From Histograms to Curves
- Define the Probability Density Function (PDF), $f(x)$, and verify its two key properties: $f(x) \ge 0$ and the total area $\int_{-\infty}^{\infty} f(x) \, dx = 1$.
- To find the probability $P(a < X < b)$, calculate the definite integral of the PDF between the limits $a$ and $b$: $\int_{a}^{b} f(x) \, dx$.
- Calculate the expectation (mean) using $E(X) = \int_{-\infty}^{\infty} x f(x) \, dx$.
- Calculate the variance using $Var(X) = E(X^2) - [E(X)]^2$, where $E(X^2) = \int_{-\infty}^{\infty} x^2 f(x) \, dx$.