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Chapter 5 Review

  • Page ID
    4573
  • 5.1 Properties of Continuous Probability Density Functions

    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 5.21

    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.

    5.2 The Uniform Distribution

    If \(X\) has a uniform distribution where \(a < x < b\) or \(a \leq x \leq b\), then \(X\) takes on values between \(a\) and \(b\) (may include \(a\) and \(b\)). All values \(x\) are equally likely. We write \(X \sim U(a, b)\). The mean of \(X\) is \(\mu=\frac{a+b}{2}\). The standard deviation of \(X\) is \(\sigma=\sqrt{\frac{(b-a)^{2}}{12}}\). The probability density function of \(X\) is \(f(x)=\frac{1}{b-a}\) for \(a \leq x \leq b\). The cumulative distribution function of \(X\) is \(P(X \leq x)=\frac{x-a}{b-a}\). \(X\) is continuous.

    The graph shows a rectangle with total area equal to 1. The rectangle extends from x = a to x = b on the x-axis and has a height of 1/(b-a).

    Figure 5.22

    The probability \(P(c < X < d)\) may be found by computing the area under \(f(x)\), between \(c\) and \(d\). Since the corresponding area is a rectangle, the area may be found simply by multiplying the width and the height.

    5.3 The Exponential Distribution

    If \(X\) has an exponential distribution with mean \(\mu\), then the decay parameter is \(m=\frac{1}{\mu}\). The probability density function of \(X\) is \(f(x) = me^{-mx}\) (or equivalently \(f(x)=\frac{1}{\mu} e^{-x / \mu}\). The cumulative distribution function of \(X\) is \(P(X \leq x)=1-e^{-m x}\).