Math 110.107, Calculus II (Biological and Social Sciences)

Fall 2010 Course Lecture Synopses

 

Week 13:  November 22

 

http://www.mathematics.jhu.edu/brown/courses/f10/107.htm

 

 

 

 

·       Monday, November 22:  Today, I finished with Section 12.4 and discrete random variables.  I defined the variance of a distribution, and related that to the mean.  In detail, I noted that the mean or expected value  of a random variable  is a measure of the central tendency of the distribution.  But two or more random variables can have the same mean, and yet be wildly different as distributions.  Hence more information is needed to discern two random variables.  The variance of , defined as  is the average of the squared differences of each value of  from the mean  (again weighted by their probabilities).  It is a measure of dispersion and gives information on how spread out the distribution values are.  Really there is little more I can say about this since we need to get to continuous distributions now.  The rest of the section is good stuff, but we will not be devoting time to it.  I then defined a continuous random variable, and spent much time focusing on how one goes from a discrete random variable to the continuous case.  By relying on the cumulative distribution function, one can see how to generalize.  Take a sample population and measure their height only in feet.  One can use the total number in each foot category over the total number in the population to get a probability distribution.  If one were then to repeat the measurements, but this time allowing for each inch category between feet, the probabilities would be much smaller since there are more boxes.  And as the accuracy of the measurement became perfect, the number of people in each vanishingly small bracket would go to 0.  However, if we increased the sample size at each time we increased the measurement accuracy, we could maintain a probability mass function that is reasonably the same as that of the original set of measurements, but with lots more bars in the bar chart.  Passing to infinite accuracy (and infinite population size), we would get a continuous distribution with a continuous probability mass function.  A continuous probability mass functions is called a probability density function.  To see this differently,  take any continuous cumulative distribution function .  It can be written as the anti-derivative of another function  as long as  satisfies certain characteristics:  it must have , (reall, also  also) and be always non-negative.  Then this kind of  is always a probability density function.  Notice that in this continuous set up, we still get  as the cumulative area under the curve  from  to , just as in the discrete case.  But here we also get .  What this means is that with infinite precision, it is impossible to get a perfect measurement.  Hence it makes more sense to talk about a measurement in an interval, like  or .  Indeed, with integrals,  and .  Hence .  We will continue next time.