Calculating probability from a p.d.f.
Calculating probability from a p.d.f.
Understanding Probability Density Functions
- A Probability Density Function (p.d.f) is a function that describes the likelihood of a random variable taking on a particular value.
- Unlike a Probability Mass Function (p.m.f) which is used for discrete random variables, a p.d.f is used for continuous random variables.
- The area under the curve of a p.d.f from a to b gives the probability that the random variable is between a and b.
Calculating Probabilities from a p.d.f
- The integral of a p.d.f over an interval gives the probability that the random variable takes a value from the interval.
- In other words, P(a ≤ X ≤ b) = ∫(a to b) f(x) dx, where f(x) is the p.d.f and dx is an infinitesimal slice of the x-axis.
- The entire area under the curve of a p.d.f is 1, representing the total probability of all outcomes.
Characteristics of a p.d.f
- The function f(x) ≥ 0 for all x. The p.d.f cannot take on negative values as probabilities cannot be negative.
- The total probability over all possible outcomes equals 1, therefore, ∫(−∞ to ∞) f(x) dx = 1.
- A p.d.f does not provide the probability of a single point. Instead, it provides the density of probability around a certain point.
Working with p.d.f
- The expected value (mean) of a p.d.f can be calculated as E(X) = ∫(−∞ to ∞) x * f(x) dx.
- The variance can be computed as Var(X) = E[(X − E[X])^2], which can simplify to Var(X) = E[X^2] − (E[X])^2.
- To find the median, compute the integral from negative infinity to m of the p.d.f, and set that equal to 0.5. Solve for m. This gives the value ‘m’ for which half the probability lies to its left.
Common Probability Density Functions
- Common p.d.f include the Uniform Distribution, Normal (or Gaussian) Distribution, and Exponential Distribution among others.
- Each of these p.d.f has its own parameters and formulas, but the basic concepts of calculating probabilities, expectations, and variances remain consistent.