Random Variables and Probability Models Random variables Discrete random variables Probability distribution functions Expected values variances and standard deviations The binomial distribution Continuous random variables The normal probability distributions The normal approximation to binomial Discrete Random Variable A random variable is a numerical measurement of the outcome of a random process A discrete random variable is a variable that can assume only a countable finite number of values Many possible outcomes number of complaints per day number of TV s in a household number of rings before the phone is answered number of heads in 100 coin tosses Continuous Random Variable A continuous random variable is a variable that can assume any value on an interval can assume an uncountable number of values thickness of an item time required to complete a task temperature of a solution height in inches These can potentially take on any value depending only on the ability to measure accurately How can we summarize possible outcomes and their probabilities Flip a coin 3 times The outcomes are HHH HHT HTH HTT THH THT TTH TTT X number of Heads P X 3 1 8 P X 2 3 8 P X 1 3 8 P X 0 1 8 Probability distribution function Pdf specifies the possible values a random variable can take on and their probabilities k 0 1 2 3 P X k 1 8 3 8 3 8 1 8 1 Prob s add up to 1 2 0 P X k 1 for every k 3 Other s have prob 0 What is the chance of seeing less than 2 heads Answer P X 2 P X 0 P X 1 1 8 3 8 1 2 Roll Two Dice 1 3 1 2 2 3 2 2 1 1 2 1 1 4 2 4 1 5 2 5 1 6 2 6 3 1 3 2 3 3 3 4 3 5 3 6 4 1 4 2 4 3 4 4 4 5 4 6 5 1 5 2 5 3 5 4 5 5 5 6 6 1 6 2 6 3 6 4 6 5 6 6 Outcomes pdf k 2 3 4 P X k 1 36 1 18 1 12 5 1 9 6 5 36 7 1 6 8 5 36 9 1 9 10 11 12 1 12 1 18 1 36 Expected value of a discrete random variable Mean of X over the long run The expected value of X is found by multiplying each possible value of X by its probability and then adding the products i e E X k P X k Sum over all possible values of k Examples k P X k 2 1 3 5 2 3 E X 2 P X 2 5 P X 5 2 1 3 5 2 3 2 3 10 3 12 3 4 Rolling 2 Dice k 2 3 4 P X k 1 36 1 18 1 12 5 1 9 6 5 36 7 1 6 8 5 36 9 1 9 10 11 12 1 12 1 18 1 36 E X 2 36 3 18 4 12 5 9 30 36 7 6 40 36 9 9 10 12 11 18 12 36 E X 7 By investing in a particular stock a person can make a profit in one year of 4000 with probability 0 3 or take a loss of 1000 with probability 0 7 What is this person s expected gain Pdf k P X k 4000 0 3 1000 0 7 E X 4000 0 3 1000 0 7 500 Roulette 12 Suppose you bet 1 on red If it is red you win 1 otherwise you lose 1 What is the expected gain loss k P X k 1 18 38 1 20 38 E X 1 18 38 1 20 38 2 38 1 19 Greek Lesson E X Greek letter mu for mean Population mean 2 Var X little sigma for standard deviation Population variance Variance of a random variable The variance and the standard deviation are the measures of spread that accompany the choice of the mean to measure center The variance 2 of a random variable is a weighted average of the squared deviations X 2 of the variable X from its mean Each outcome is weighted by its probability in order to take into account outcomes that are not equally likely The larger the variance of X the more scattered the values of X on average The positive square root of the variance gives the standard deviation of X Variance of a discrete random variable For a discrete random variable X with probability distribution and mean X the variance 2 of X is found by multiplying each squared deviation of X by its probability and then adding all the products A basketball player shoots three free throws The random variable X is the number of baskets successfully made X 1 5 The variance 2 of X is Value of X 0 1 2 3 Probability 1 8 3 8 3 8 1 8 2 1 8 0 1 5 2 3 8 1 1 5 2 3 8 2 1 5 2 1 8 3 1 5 2 2 1 8 9 4 2 3 8 1 4 24 32 3 4 75 Measuring Spread In general Var X E X 2 E X2 2 Expected squared distance from mean k 2 k k P X k 1 1 1 2 0 0 2 6 1 1 3 2 E X 2 1 2 2 2 2 2 2 6 3 2 2 2 1 2 0 6 1 2 4 k P X k 2 25 Example 0 5 1 25 E X 5 0 25 25 E X2 4 25 0 25 1 25 Var X 1 25 0 0625 1 1875 Rules for means and variances If X is a random variable and a and b are fixed numbers then E a bX a bE X Var a bX b2Var X If X and Y are two independent random variables then E X Y E X E Y Var X Y Var X Var Y The standard deviation of a random variable is SD X Var x Example The expected annual payout per insurance policy is 200 and the variance is 14 960 000 If the payout amounts are doubled what are the new expected value and variance E X 2 2 2 Var X 2 2 200 400 E X 4 14 960 000 59 840 000 Var X 2 Compare this to the expected value and variance on two independent policies at the original payout amount E X Y E X Var Y Var X 2 200 400 E Y 2 14 960 000 29 920 00 Var X Y Note The expected values are the same but the variances are different 2011 Pearson Education Inc Investment You invest 20 of your funds in Treasury bills and 80 in an index fund that represents all U S common stocks Your rate of return over time is proportional to that of …
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