Calculating Probabilities with a continuous random variable

A continuous random variable is one which can take an infinite number of possible values. This is in contrast to a discrete random variable, which can only take distinct, separate values.
The probability density function (pdf) of a continuous random variable signifies how probabilities are distributed across the range of possible values.
Unlike discrete random variables, where individual outcomes have specific probabilities, the probability of a continuous random variable taking any exact value is always zero because there are infinite possible values it can take.

To calculate probabilities with a continuous random variable, you’ll utilise the probability density function (pdf) and integration.
The probability of a continuous random variable taking on a value within a certain range is given by the integral of the pdf over that range. In other words, you’re considering the area under the curve of the pdf over that interval.
The definite integral from ‘a’ to ‘b’ of the pdf, denoted as ∫f(x)dx from a to b, is equal to the probability that the random variable takes a value within the interval [a, b].
The total probability under the curve of a pdf for a continuous random variable is 1, reflecting that one of the possible outcomes must occur. This is represented as ∫f(x)dx from -∞ to ∞ = 1.

The expected value (or mean) of a continuous random variable is calculated as the definite integral of x times the pdf over all possible values of x.
The variance of a continuous random variable is calculated as the definite integral of (x - expected value)^2 times the pdf over all possible values of x.

Remember that the probability of a continuous random variable taking any exact value is zero.
Calculation of probabilities for continuous random variables uses integration rather than simple multiplication and addition.
The area under the curve of the pdf over an interval gives the probability that the random variable takes a value within that range.