Introduction To Probability Models
Introduction To Probability Models STAT 22500
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Stat 225 Topic 4 Random Variables Random Variables lfthe probability consisted only of the methods described in the previous section there would be little reason to take this class We would also live in a very different world one without many of the analyses that have become a standard part of our lives weather prediction market prediction experimental design medical diagnostics and many others rely upon probability However there is something missing a step that allows us to make the leap from calculating probabilities to using them in many of these applications To demonstrate this consider the following questions we might ask about the results of a following random experiment Which if any can we answer using the previous techniques 0 How often do we expect a particular event to occur 0 What events do we expect to happen on average How much will knowing the average help us make guesses about the results of a random experiment ls our actual result generally going to be near the average77 How can we judge whether or not it is a good idea to perform a random experiment in which risk is involved Think of random experiments that involve bets of money or the potential of injury etc Can you think of any other limitations imposed by our previous method of calculating probability These latter ideas require a big change in how we look at probability The rst step is by using random variables De nition A random variables is a whose domain is a How Random Variables Work Random variables are a way of simplifying experiments lt7s often true that we dont need to know every detail about what happens in a random experiment Instead we want to summarize it with a small amount of relevant information For example 0 The sum of two dice but not the exact roll 0 The amount of money we win or lose by playing a game but not the precise winning conditions 0 Your overall score on the next Stat 225 exam but not how you achieved those points random variable takes each potential outcome and translates it into a number For example suppose X is a random variable representing the sum of two dice X 6 can be achieved from the outcomes X 3 can be achieved from the outcomes A random variable takes on some numerical value for every possible outcome in our sample space Example Toss a fair coin three times Suppose we are interested in the random variable X number of heads The following graph matches each possible element in the sample space to the corresponding value of X Example Roll 5 Gsided dice and let X be the sum of the dice What is X532 X142 X335 Types of Random Variables There are two fundamentally di erent kinds of random Variables random Variables take on a nite or countably in nite number of values These variables will usually have integer values althou h sometimes they will include fractions These random Variables will be the focus of the course until the next midterm random variables take on a continuum or range of values We will study these random variables after the next exam Example Are the following random Variables discrete or continuous 1 The number of MampM s in jar 2 The top wind speed recorded by the Purdue Airport during February 3 You re nal grade in Stat 225 Working with Random Variables Notation Random variables are always written using letters usually toward the back of the alphabet The value that a random variable takes on is written using letters usually corresponding to the letter describing the random variable Numerical Nature lf you7re trying to come up with a random variable to describe a discrete sample space try asking a question that starts with how many For example suppose we have a box with 7 red balls and 8 yellow balls What is a potential random variable that might be of interest Relationship with Events We can describe events using random variables Suppose we let A be the event in which X 6 All outcomes in which X 6 are elements in the set A We dont even have to name the events often well just say the event in which X 677 Mutual Exclusivity The events X x and X y where z 31 y are always mutually exclusive This is because the same outcome cannot have two different values of X at the same time Thus if we are interested in the probability X x or X y we can Probability Mass Functions A probability mass function is For X number of heads in three rolls of a fair coin7 ll in the following table I WW Probability mass functions can also be displayed as a graph Basic Properties of PMFS All PMFs must satisfy three properties Compare these to the axioms of probability discussed on page 9 of Topic 1 What similarities do you see 1 Computing Probability with PMFs Soon7 we will learn ways to utilize patterns in order to compute PMFs very quickly For right now7 we can still utilize the same methods we7ve already learned Example In three tosses of a fair coin7 what is the probability that you get exactly two heads Fundamental Probability Formula Sometimes we7re interested in the values of X that fall into a range For example7 we might want 0 S X S 2 How would you calculate P0 S X S 2 in the previous example In general Example A box contains one red7 two orange7 and ve black balls You choose two balls at random without replacement For every red ball you win 57 for every orange ball you win 17 and you do not win anything for a black ball Let X be the amount of money you7ll win a Write down the probability mass function of X b What is the probability that you will win 3 or more c Assuming you win something7 what is the probability that you win 3 or more Expected Value and Variance The biggest advantage of using random variables is that rather than looking at a number of events individually AB7 and C for example7 we can study how changing the value of a random variable1 a 2 a 3 changes the probability This goes beyond calculating probabilities to helping to predict what might happen in the real world Consider the following example In a simple game7 two fair coins are tossed and the payoff is to be determined from the outcome You may choose one of the following payoff strategies Payoff 1 Win 1 for each head Lose 2 for two tails Payoff 2 Win 1 if the coins are both tails Win 2 if the coins are different Lose 2 if the coins are both heads Let X denote your total winnings if you play the game once Write down the probability mass function for X for each strategy What are some criteria you might use to judge each game Which payoff strategy would you rather play Why or why not Expected Value Suppose I offer you the following game I ip a coin then you ip two coin If both of your coins match mine you win 2 Otherwise you lose 1 Would you play Two factors should contribute to your decision as to whether or not to play this game First what is the probability of winning Second how much stands to be gained or lost in the case of a win or a loss Putting these questions together you need to nd out whether or not you expect to gain more than you lost in the process of playing this game In probability this is called the expected value De nition The expected value of a random variable X with PMF pXz is given by Note We may interchangeably use the terms mean average expectation and expected value and the notations EX0ru The expected value is a weighted average of the possible values of X weighted by the probabilities Remark The expected value of a discrete random variable is the long run average value of a random variable when repeating the same experiment very many times In the long run you expect the average of the random variable X to be close to the expectation Center of Gravity EX can be understood as the physical Center of Gravity77 or balance point77 in the histogram of the PMF if you treat the probability to be literally mass Example Let X be a random variable with PMF s 1 0 1 2 pXx 13 14 14 16 Then EX Fundamental ExpectedValue Formula Consider for a moment what would happen if we let Y X2 in the example above a What are the possible values Y might take on b Calculate the PMF table for Y c What is EY7 Notice the similarities between the PMF table for X and the PMF table for Y7 and the calculation of the expected value In general7 we have Practice For the previous example7 calculate the following a EEX b EX 3 c E4X 5 Expectation is 21 Linear Operator Let XLXZ7 Xn be random variables7 and let ab be constants Then Example Roll two six sided dice Let X be the number that comes up on the rst dlie7 let Y be the number that comes up on the second die Calculate the following a EX c EX Y d EX 7 3y Variance Suppose I offer you the following two investments a You invest 1 and will gain 9 with probability 01 b You invest 1 and will gain 999999 with probability 0000001 Calculate the expected value for both investments Which game do you prefer Do you think there is a right or wrong answer to this question Why or why not De nition Let X be a discrete random variable with probability mass function Then the variance of X is de ned as Remarks 0 The variance is sometimes abbreviated as 02 o VarX is always non negative VarX 2 0 o The variance is a measure of how spread out the variables are If VarX 07 then the spread is zero7 ie all the probability is concentrated in one point no randomness o The variance is not measured in the same units as the random variable It is measured in units A histogram can give a good picture as to what the variance is calculating Example cont Let X be your gain from the two investment strategies offered above Then the probability mass functions of X under the two strategies are Compute the variance of the gain in both cases The formula for variance can be rewritten with a little algebra into a far easier form for computation VarX E X i EX2 E X2 i 2XEX EX2 EX2 i 2EXEX EX2 EX2 7 E002 On the other hand7 variance is not a linear operator Let X be a random variable and ab be constants Then VaraX b E aX b2 7 EaX b2 E aZXZ 2abX b2 i aEX bf a2EX2 2abEX b2 i a2EX2 i 2abEX i b2 a2 EX2 i EX2 Standard Deviation Remember that the variance of a random variable X does not have the same units as the random variable itself Its not very dif cult to convert units though we just need to take the of the variance This gives the standard deviation Examples 1 Let X be a discrete random variable with PMF z 0 1 2 3 pXz 010 020 040 030 a Find EX b Find VarX 0 Find Var2X 7 5 2 Suppose X is a random variable with M 2 and 02 9 a Find EX2 b Find EX 7 1 0 Find the standard deviation of X Extra Practice pmlemeu pleyedpypleemgmemmmepems e39eme keuuemepem Omeel ere meae ewuee1 wnh ax equeuy sued end uumpeueu em 1 Spun end esmen pen 1 msde to mdA me em Whm the whee1 stops spinnmg me he 1 euueny Melv to lsnd In one 0 me 51m K your hem Includes the uumpeu d me 5111 wnh me hell you w Wm money epemmgm whet type dbem you have meae Complete me Mowing 39eme A ueeenpum deeeupemeyeuememmeumpeea Bea p smug Bot 35 1 ZrNumhm Bea 17 1 Raw Bot 11 1 ArNumhm Bea a 1 ErNumhm Bea s 1 lzr mhm 2 1 Bot mrNumhm 1 1 Bot Nate an xeadmg th table A 1 1 payout k1 every 1 you heme ewm what your aligns ddlsx hack Plus enmheu ddlsx A as 1 payout means that k1 every 1 you pen ewm wul yve you your meme dn ex hack Plus 35 mae ddlsxs 1a Straight Bet A straight bet is a bet on a single number The chips being bet are placed within the square corresponding to the number on the board 2Number39 Bet A 2 number bet is a bet that one of two adjacent numbers will win The chips being bet are placed on the line between the two numbers on the board Row Bet A row bet is a bet on a particular row of three numbers Its also possible to bet on 3 number combinations of 0 00 17 27 and 3 provided that there is a corner to put the chips that touches all 3 desired numbers To place a row bet7 the chips bet are placed on the line to the outside of the row For example7 to place a row bet on 1 2 37 the chips would be placed on the line between 3 and the outside of the board 4Number39 Bet A 4 number bet is a bet on any of four numbers forming a square on the board The chips are placed on the corner touching all four desired numbers 6Number39 Bet A 6 number bet is a bet placed on the numbers in two adjacent columns The chips are placed on corner between the two desired columns7 to the outside of board 12Number Bet A 12 number bet can be placed on a single column7 on the rst 12 num bers7 the 2nd twelve numbers7 or the third twelve numbers There are marked sections of the board on which these bets are placed 18Number Bet An 18 number bet can be placed on odd7 even7 red7 black7 1 187 or 19 36 There are marked sections of the board to place each of these bets