ClassNotes-Chapter-06

ClassNotes-Chapter-06 - Chapter 6: Continuous probability...

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1 © Sven Thommesen 2011 Chapter 6: Continuous probability distributions [Edited 10/05/08] Recall from Chapter 5 that we used the probability mass function f(x i ) to assign a probability Pr(x=x i ) to every possible outcome x i of a discrete random variable X. We could do this because the number of possible outcomes was limited. A discrete probability function is a probability function which assigns probabilities to possible values of a discrete variable “x”. That is, our experiment produces outcomes that are whole numbers (integers), and the pmf assigns a probability value to every possible integer outcome. A discrete pmf obeys the general rules of probability this way: the probability of any specific outcome must be between 0 and 1 the sum of probabilities for all possible values of “x” (all elements in the sample space) must be equal to 1. the probability of any “x” that is not in the sample space is 0. For the pmf, we write: Pr(x = x 0 ) = f(x 0 ) We have discussed different examples of discrete pmf’s: the discrete uniform probability function the binomial probability function the Poisson probability function the hypergeometric probability function … and there are many more, each suited for use in some particular set of circumstances. [For statisticians, part of the job is to determine which pmf to use for a given problem.]
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2 PDF: the Probability Density Function A continuous probability function is a probability function which assigns probabilities to possible values of a continuous (real) variable “x” . When the variable x is a continuous variable (consisting of real numbers), assigning probability to a specific value of x is no longer possible, since between any two real numbers (say, 1.0 and 2.0) there is literally an infinite set of additional real numbers! For the simplest case of a uniform probability function, the probability of a given value for x would then have to be P (x) = 1/N = 1/∞ = 0! So for continuous probability functions we need to interpret things a little differently. For continuous variables, we do not talk about the probability that x will be equal to a specific value x 0 (always equal to zero), but rather about the probability that x will fall in the interval between two values x 0 and x 1 , written Pr(x 0 ≤ x ≤ x 1 ) We still draw a probability function, here called the probability density function or PDF, corresponding to our relative frequency graphs in chapters 3 and 5. [Diagrams: a bell curve with limits x 0 and x 1; the Exponential pdf] In the diagrams above, we interpret the probability of the event “x falls between x 0 and x 1 ” as the area under the probability density curve between x 0 and x 1 . Note that the sample space for a continuous probability function must be an interval on the real line: it can be (- ∞, +∞), or [0, +∞), or [a,b] for arbitrary values a and b.
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ClassNotes-Chapter-06 - Chapter 6: Continuous probability...

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