121616949-math.229 - called the mean is generally most...

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
9.8 Probability 215 An even function is one that is symmetric around the y axis. EXAMPLE 9.31 If f is an even probability density function then one of the medians is 0. If f ( x ) > 0 on [ - δ, δ ] for some δ > 0, then 0 is the only median. In particular, the median of the standard normal distribution is 0. DEFINITION 9.32 If a probability density function f has a global maximum at x = c then c is a mode of the random variable X . There need not be a single mode; for example, in the uniform distribution every number between a and b is a mode. The mode may not even exist, though it is a bit tricky to come up with an example. EXAMPLE 9.33 It is somewhat difficult to devise a continuous probability density function for which there is no mode, but easy if we give up continuity. Consider f ( x ) = 0 x < - 1 x + 1 - 1 x < 0 0 x = 0 - x + 1 0 < x 1 0 1 < x . Sketch this graph; it is then apparent that f has no global maximum and hence no mode. In practice, both the mode and the median are useful, but the expected value, also
Image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: called the mean, is generally most useful. Following our discussion of discrete probability, the definition should not be surprising. DEFINITION 9.34 The mean of a continuous random variable X with probability density function f is μ = E ( X ) = Z ∞-∞ xf ( x ) dx , provided the integral converges. When the mean exists it is unique, since it is the result of an explicit calculation. The mean does not always exist. Let us look more closely at the definition of the mean. It is in fact essentially identical to the definition of the center of mass of a one-dimensional beam. The probability density function f plays the role of the physical density function, but now the “beam” has infinite length. If we consider only a finite portion of the beam, say between a and b , then the...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern