Continuous Probability Distributions

Probability comes from area

For a continuous model with density f(x), the probability of a range is an integral:

$P(a\le X\le b)=\int_a^b f(x)\,dx=F(b)-F(a).$

The density is nonnegative and its total area is 1. Its height is not a probability at a point and may exceed 1. For an absolutely continuous model, changing an endpoint from < to ≤ does not change the probability because an exact point has probability 0.

Record parameter conventions. One reference may use an exponential rate λ; another uses scale β = 1/λ. Mixing them changes the calculation.

Worked example: uniform waiting time

Suppose a hypothetical shuttle arrives exactly every ten minutes and a passenger arrives uniformly within a cycle. Model the wait W as Uniform(0,10). The density is 1/10 per minute on that interval.

P(2 ≤ W ≤ 5) = (5 − 2)/10 = 0.3. The expected wait is (0 + 10)/2 = 5 minutes, and the variance is (10 − 0)²/12 = 8.3333 minutes².

Uniform arrival within a perfectly regular schedule is an assumption. A timetable-aware passenger or irregular shuttle intervals can invalidate this model. A uniform distribution is not a default for everything bounded.

Worked example: exponential waiting time

A constant-rate Poisson process with rate λ = 0.2 events per minute has exponentially distributed time to the next event. Its mean wait is 1/λ = 5 minutes, its standard deviation is also 5, and:

$P(W>t)=e^{-\lambda t}.$

P(W > 10) = e⁻² ≈ 0.1353. Its median solves e⁻⁰·²ᵗ = 0.5, giving ln(2)/0.2 ≈ 3.4657 minutes. This is a different distribution from the uniform example even though both means are 5.

The exponential model is memoryless: P(W > s + t | W > s) = P(W > t). Waiting longer does not make arrival due under this model. Many real waiting and lifetime processes have changing rates and are not memoryless.

Other models answer other questions

A normal distribution describes a symmetric unbounded model using mean and standard deviation. A gamma distribution can model positive, right-skewed outcomes; its shape and scale conventions need attention. A beta distribution lies between 0 and 1 and can model uncertainty about a probability.

The t, chi-square and F distributions arise as reference distributions of certain statistics under assumptions. Their use does not imply that the raw observations follow those distributions.

Assess support, units, shape, measurement process and tails. A fitted distribution is a model to evaluate, not an automatically true account of the population.

Practice

For Uniform(0,4), is a density of 0.25 the probability of observing exactly 2?

Show the explanation

No. An exact point has probability 0 under the continuous model. The probability of [1,2] is its width 1 multiplied by density 0.25, giving 0.25.

Continue

Study the normal distribution. NIST provides uniform and exponential reference formulas; the exponential page uses scale rather than the rate notation used here.