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by: Reyes Glover


Reyes Glover
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Class Notes
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This 4 page Class Notes was uploaded by Reyes Glover on Sunday September 6, 2015. The Class Notes belongs to M 358K at University of Texas at Austin taught by Staff in Fall. Since its upload, it has received 10 views. For similar materials see /class/181523/m-358k-university-of-texas-at-austin in Mathematics (M) at University of Texas at Austin.

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Date Created: 09/06/15
M 358K Spring 06 REVIEW AND MORE OF RANDOM VARIABLES In calculus you learned things like If an object is tossed straight up from initial height he with initial velocity v0 then its height at time tseconds after being released is ht 2 ho vatgtz If you read the textbook carefully andor listened carefully in lecture you might have heard a qualification like quotif you ignore air resistance quot In reality air resistance can be important in most cases it results in a quotterminal velocityquot which the object cannot exceed However finding an equation of motion taking air resistance into account presents real problems since the air resistance depends on the mass and also the shape of the object And if the object is a parachutist its shape may be changing as it falls In fact can you really know the initial velocity exactly The initial height And what if there s wind As this example shows in real life we often don39t have deterministic that is exact formulas like ht h0 Vat gt2 Instead we have to deal with approximations and uncertainty So we need stochastic that is probabilistic methods A key idea in probabilistic methods is the idea of random variable The height of our real falling object can be considered as a random variable we may be able to find a formula taking into account air resistance that will give an approximate description of the object s motion but there are still uncertain factors such as wind and our ability to measure the initial velocity or determine the effect of air resistance exactly Your probability textbook may have defined a random variable as A real valued function defined on a sample space This is technically correct but what is often more helpful in statistics is to think of a random variable as a variable that depends on a random process Here are some examples 1 Toss a die and look at what number is on the side that lands up Tossing the die is an example of a random process the number on top is the random variable 2 Toss two dice and take the sum of the numbers that land up Tossing the dice is the random process the sum is the random variable 3 Toss two dice and take the product of the numbers that land up Tossing the dice is the random process the product is the random variable Examples 2 and 3 together show that the same random process can be involved in two different random variables 4 Randomly pick a UT student and measure their height Picking the student is the random process their height is the random variable 5 Randomly pick a student in this class and measure their height Picking the student is the random process their height is the random variable Examples 4 and 5 illustrate that using the same variable height but different random processes gives different random variables 6 Measure the height of the third student who walks into this class What is the random process In Example 5 the random process was done deliberately in Example 6 the random process is one that occurs naturally Can you explain how the di erent random processes make these two random variables di erent 7 Toss a coin and see whether it comes up heads or tails Tossing the coin is the random process the variable is heads or tails This example shows that a random variable doesn39t necessarily have to take on numerical values 8 The time it takes for an IF shuttle bus to get from 4 h and Speedway to the Dean Keaton stop is a random variable Whoa you may say where39s the random process I39ve given this example this way precisely because random variables are often detined in this way The random process here is quotimplicitquot at least tor those used to defining random variables in this way What is really meant is quotHandoml y pick an IF shuttle bus run and measure the time it takes to get from 4539 and Speedway to the Dean Keaton stop quotSo the random process is picking the shuttle bus run and the random variable is the time measured 9 a The height t minutes after its release of an object tossed straight up from initial height h0 with initial velocity V0 b The height we measure t minutes after its release of an object tossed straight up from initial height h0 with initial velocity V0 c The formula we obtain taking everything we can into account for the height t minutes after its release of an object tossed straight up from initial height h0 with initial velocity V0 How are these three random variables di erent What is the random process involved in each Hints They re complex CAUTION If you look in a dictionary you may find that the first definition of random is something like Having no specific pattern or objective haphazardquot This is NOT the technical meaning of random that is used in probability and statistics Here are some examples of processes that are random in the technical sense A We consider a process such a tossing a die or a coin to be random B When we talk about randomly picking a UT student or randomly picking an IF shuttle run we mean using a process that gives each possible UT student or IF shuttle run an equal chance of being chosen We can imagine but only imagine for example numbering all UT students 1 2 3 etc and having a huge die with as many sides as there are UT students Tossing the die and taking the student whose number came up would be a way of randomly picking a UT student In practice random selections such as this are made by replacing our imaginary huge die by a computer program called a pseudo random number generator or random number generator for short that is designed to give essentially the same result These examples however might lead one to believe that a random process always has to give every outcome and equal chance of happening This is not the case We could imagine for example a quotloadedquot or quotbiasedquot die that was made so that one of the sides came up more frequently than the others We would still consider tossing this die to be a random process One important aspect of a random process is that although there may be and usually is a pattern in the long run there is no way of knowing in advance the result of one occurrence of the process In other words the result of one occurrence of a random process is uncertain but we can at least in theory say something about the longterm behavior that is what happens over many many occurrences of the process Examples of random variables that people are interested in studying include the following using the convention mentioned in example 8 39 The time to breakdown of a computer 39 The yield per acre of a field of wheat 39 The birth weight of a child 39 The length of time a person lives 39 The Dow Jones Index 39 The concentration of ozone in the air 39 How much time is required to read a disk sector into main memory As you have seen in Probability and in Section 11 of the M 358K textbook one way to give information about that is to say something about the long term behavior of a random variable is to show its distribution In the vocabulary of probability this is the graph of the probability density function In Probability you usually started with a formula for the probability density function and got the distribution from that In this course we will often use empirical distributions That means that they are based on actual data plotted in a histogram or other graph such as a stem plot Example Here is a histogram that gives the empirical distribution of the heights of students in a particular beginning statistics class Figure 1 The first bar shows the number of people whose height is between 58 and 595 inches inclusive the second bar the number of people whose height is between 60 and 615 inches and so on Please note There are lots of different conventions for drawing histograms so please use caution in interpreting them Based on this histogram and whatyou know about peoples39 heights what would you guess the proportion of males and females in this class to be How would you expect the histogram to differ if that proportion were di erent The histogram above is an example of a frequency histogram Sometimes we may use a density histogram 39 In a frequency histogram the bar above each interval shows the number of values in the interval 39 In a density histogram the bar above each interval shows the proportion of values in the interval Here is a density histogram for the heights of students in the same class mo Demiiy o 8 I Figure 2 What is the same about the two histograms Why


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