A continuous random variable

A characteristic of continuous random variables is that the set of their possible values is uncountably infinite (a union of intervals on the real line), although the range of their values can be described by a function.

The values of continuous random variables are physical values from the real world, such as:

• the height of a student in the 5th grade of elementary school,
• the mass of a pile of potatoes,
• the time interval between the passage of two cars on a highway, etc.

With such variables, the number of different values that a random variable can take is limited only by the precision of the measuring instrument used to obtain those values.

If the limitations of the measuring instrument are too great, so that the number of possible values of the random variable is relatively small, it is possible to treat a continuous variable as a discrete one and vice versa; if the number of possible values that a discrete random variable can take is relatively large, it is possible to treat a discrete random variable as continuous.

Since the number of possible values of a random variable is infinite, the probability that a continuous random variable takes on exactly one specific value is 0, i.e., P(X = x) = 0. Therefore, the probability that a random variable is in a certain range is always calculated.

The distribution of a continuous random variable is usually represented graphically by a probability density curve.

The probability that the value of a random variable lies between any two points X₁ and X₂ plotted on the X axis is equal to the area of the diagram enclosed by the X axis, the probability density function curve, and a line drawn normal to the X axis from the points X₁ and X₂ to the probability density function curve.

The probability density function is non-negative, since probabilities cannot be negative values.

Since the probability is represented as the area under the curve, it is calculated as a definite integral

The total area under the probability density function curve is 1.

If the probability density function f has a unique maximum at a point x_m, then that point x_m is the mode of the distribution.

For the probability density function, it is possible to calculate measures of central tendency and measures of variability, as well as for the probability function of a discrete variable, and they represent the parameters of the distribution.