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4.1: Probability Density Functions (PDFs)

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  • Recall that continuous random variables have uncountably many possible values (think of intervals of real numbers). Just as for discrete random variables, we can talk about probabilities for continuous random variables using density functions.


    The probability density function (pdf), denoted \(f\), of a continuous random variable \(X\) satisfies the following:

    1. \(f(x) \geq 0\), for all \(x\in\mathbb{R}\)
    2. \(f\) is piecewise continuous
    3. \(\displaystyle{\int\limits^{\infty}_{-\infty}\! f(x)\,dx = 1}\)
    4. \(\displaystyle{P(a\leq X\leq b) = \int\limits^a_b\! f(x)\,dx}\)

    Example \(\PageIndex{1}\):

    Let the random variable \(X\) denote the time a person waits for an elevator to arrive. Suppose the longest one would need to wait for the elevator is 2 minutes, so that the possible values of \(X\) (in minutes) are given by the interval \([0,2]\). A possible pdf for \(X\) is given by
    $$f(x) = \left\{\begin{array}{l l}
    x, & \text{for}\ 0\leq x\leq 1 \\
    2-x, & \text{for}\ 1< x\leq 2 \\
    0, & \text{otherwise}
    The graph of \(f\) is given below. The reader is encouraged to verify that \(f\) satisfies the first three conditions in Definition 4.1.1.


    Figure 1: Graph of f

    So, if we wish to calculate the probability that a person waits less than 30 seconds (or 0.5 minutes) for the elevator to arrive, then we calculate the following probability using the pdf and the fourth property in Definition 4.1.1:
    $$P(0\leq X\leq 0.5) = \int\limits^{0.5}_0\! f(x)\,dx = \int\limits^{0.5}_0\! x\,dx = 0.125\notag$$