Probability Distributions Continuous Random Variables

 Continuous Random Variable: Can assume a whole range of values which cannot be listed: Examples:  Arrival times of customers using an ATM at the start of the hour  Time between arrivals  Length of service at an ATM machine  Used in our Project

Continuous Random Variables  A Thought Experiment  Suppose a car is moving with increasing velocity according to the table shown below:  How far has the car traveled?

Continuous Random Variables  During first two seconds the car is probably moving at a faster & faster speed.  But at the very minimum of 20 ft/sec  At the very minimum, the car would have to have traveled 20 ft/sec x 2 sec or 40 ft

Continuous Random Variables  If we look at the next two seconds:  Car moves at 30 ft/sec (smallest possible speed)  So, smallest possible distance traveled during this time: 60 ft

Continuous Random Variables  If we continue in this fashion, for 10 seconds, at the very minimum:  This is the smallest possible distance traveled.  It is only an estimate

Continuous Random Variables  We can reason in a similar fashion for the maximum distance traveled:  This is the maximum distance traveled.  This is also an estimate.

Continuous Random Variables  Somewhere in between our two estimates is the actual distance traveled:  Notice that there is a difference of 60 feet between estimates.

Continuous Random Variables  The only way we can improve on our estimates is to record the velocity more frequently:

Continuous Random Variables  Proceeding as we did when the velocities were recorded every 2-seconds:

Continuous Random Variables  Now our estimates are much closer  Notice that they are only separated by 30 which is half the separation we saw in the other table  Halving the time interval, halved the difference between max and min

Continuous Random Variables  Visualizing distance traveled on graph  Area of each bar corresponds to estimated distance traveled on that time interval  Sum of all areas = estimated distance traveled (under- estimate)

Continuous Random Variables  Visualizing distance traveled on graph  Area of each bar corresponds to estimated distance traveled on that time interval  Sum of all areas = estimated distance traveled (over- estimate)

Continuous Random Variables  For 1-second intervals:  Notice that the areas are closer to estimating the actual distance traveled with more velocities being recorded. Under-estimateOver-estimate

Continuous Random Variables  For 0.5-second intervals:  Notice, the more rectangles the closer both estimates get to the actual distance traveled. Under-estimate Over-estimate

Continuous Random Variables  How else could we make our estimate more accurate? [MIDPOINTS]

Continuous Random Variables  Why are the midpoints more accurate? Amount over and the amount under are almost identical, so the area is an even better approximation to the distance traveled on that interval than the over and under estimates previously used

Continuous Random Variables  Recording the velocity more often give us a better approximation.  Midpoints do this better & faster 1 second velocities0.5 second velocities

Continuous Random Variables  If we kept making more rectangles, the upper silhouette of the histogram becomes smoother:  Actual Distance Traveled = area under velocity graph from [0,10]

Continuous Random Variables  Finding areas under graphs of function is called integral calculus  The integral of a function over some interval is the area under the function on that interval and is denoted:

Continuous Random Variables  Another Example: US Age Distribution  What percentage of the population fall between age 0 to 20?

Continuous Random Variables  Make a histogram:  Find area under “0 to 20” bar

Continuous Random Variables  The “0 to 20” bar:  Height: 1.45 %/yr Width: 20 yr  Area = 1.45 %/yr x 20 yr = 29 %  So around 29% of the population is around 0 to 20 years old.

Continuous Random Variables  Find the % of population that is between 20 & 60 years old?  Need to find area under two bars:

Continuous Random Variables  Find % of population less than 10 yrs old?  Assume that population was evenly distributed over 0 to 20 group  0 to 10 is about half of the 0 to 20 group which was 29% so about 14.5% would be a good estimate.

Continuous Random Variables  % of population between 75 and 80?  Area of “60 to 80” = 13%  “75 to 80” is about 1/4 th of the “60 to 80” group so: about 3.25%  Probably not a good estimate  Probably more 60 to 65 people than 70 to 75 (not uniform like the other age group)

Continuous Random Variables  Just like with the velocity problem, if we record ages in smaller groups we get a “smoothed out” graph:

Continuous Random Variables  The function graphed on the previous slide is the “smoothed out” histogram called the probability density function (p.d.f)  Area under f(x) from 0 to 100 is 1 (since this is the total population)

Continuous Random Variables  In fact any area under the p.d.f is telling you the probability that your continuous random variable is between those values:

Continuous Random Variables  An area:  If you want to learn how to find this area without having to make a bunch of rectangles take MATH 124

Continuous Random Variables  What would be?  Answer: ZERO! ; all probabilities under a p.d.f graph are areas. There’s no width so area is zero!

Continuous Random Variables  The results of the last slide tell us that the following are all equivalent:

Continuous Random Variables  What does the p.d.f function f(x) itself tell us?  Example: What is f(20)?  On the graph, f(20) = 1.4  But what are the units?

Continuous Random Variables  f(20) = 1.4 % per yr of age  This is NOT telling us that 1.4 % of the population is precisely 10 years old (where 10 years old means exactly 10, not 10½, not 10.1).  Only the area of some region will tell you the % of population that are in a certain age range.

Continuous Random Variables  What about the c.d.f?  Still defined as  But we need to use our area concept instead

Continuous Random Variables  The p.d.f is shown:  If we find the area starting from 0 to some point x, we’ll get a function that increases to 1 as x increases to 100

Continuous Random Variables  The graph of the c.d.f F X (x):

Continuous Random Variables  p.d.f vs c.d.f p.d.f c.d.f

Continuous Random Variables  p.d.f: areas under this curve tell you probabilities  c.d.f: the height of the function at a given x value tells you the P(X ≤ x)

Continuous Random Variables  p.d.f vs c.d.f F X (20) P(X≤20) = area under curve Both are equivalent

Continuous Random Variables  Recall:  Last slide tells us that: also area under p.d.f. graph, above x-axis, and between a and b

Continuous Random Variables  What about E(X)?  Recall that for finite random variables:  For continuous random variables:

Continuous Random Variables  The last bullet on the previous side is not easy to answer without calculus concepts  This vertical line is the expected value E(X) = 45.34

Continuous Random Variables  Geometrically, the E(X) for a continuous random variable is the point on the horizontal axis where the region under the p.d.f., if it were made out of cardboard, would balance Balancing point