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5.1: Continuous Probability Functions

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    We begin by defining a continuous probability density function. We use the function notation \(f(x)\). Intermediate algebra may have been your first formal introduction to functions. In the study of probability, the functions we study are special. We define the function \(f(x)\) so that the area between it and the x-axis is equal to a probability. Since the maximum probability is one, the maximum area is also one. For continuous probability distributions, PROBABILITY = AREA.

    The probability density function (pdf) is used to describe probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In other words, the area under the density curve between points a and b is equal to \(P(a < x < b)\). The cumulative distribution function (cdf) gives the probability as an area. If \(X\) is a continuous random variable, the probability density function (pdf), \(f(x)\), is used to draw the graph of the probability distribution. The total area under the graph of \(f(x)\) is one. The area under the graph of \(f(x)\) and between values a and b gives the probability \(P(a < x < b)\).

    The graph on the left shows a general density curve, y = f(x). The region under the curve and above the x-axis is shaded. The area of the shaded region is equal to 1. This shows that all possible outcomes are represented by the curve. The graph on the right shows the same density curve. Vertical lines x = a and x = b extend from the axis to the curve, and the area between the lines is shaded. The area of the shaded region represents the probabilit ythat a value x falls between a and b.
    Figure \(\PageIndex{8}\)

    The cumulative distribution function (cdf) of \(X\) is defined by \(P(X \leq x)\). It is a function of \(x\) that gives the probability that the random variable is less than or equal to \(x\).

    Formula Review

    Probability density function (pdf) \(f(x)\):

    • \(f(x) \geq 0\)
    • The total area under the curve \(f(x)\) is one.

    Cumulative distribution function (cdf): \(P(X \leq x)\)

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