1 1 Econ 240A Power Four 1 Last Time Probability 2 Problem 6 61 A survey of middle aged men reveals that 28 of them are balding at the crown of their head Moreover it is known that such men have an 18 probability of suffering a heart attack in the next ten years Men who are not balding in this way have an 11 probability of a heart attack Find the probability that a middle aged man will suffer a heart 3 attack in the next ten years P Bald and MA 0 28 Bald Not Bald Middle Aged men 4 Bald Not Bald Middle Aged men 5 P Bald and MA 0 28 P HA Bald and MA 0 18 P HA Not Bald and MA 0 11 Probability of a heart attack in the next ten years P HA P HA and Bald and MA P HA and Not Bald and MA P HA P HA Bald and MA P BALD and MA P HA Not BALD and MA P Not Bald and MA P HA 0 18 0 28 0 11 0 72 0 054 0792 0 1296 6 Random Variables There is a natural transition or easy segue from our discussion of probability and Bernoulli trials last time to random variables Define k to be the random variable of heads in 1 flip 2 flips or n flips of a coin We can find the probability that k 0 or k n by brute force using probability trees We can find the histogram for k its central tendency and its dispersion 7 Outline Random Variables Bernoulli Trials example one flip of a coin expected value of the number of heads variance in the number of heads example two flips of a coin a fair coin frequency distribution of the number of heads one flip two flips 8 Outline Cont Three flips of a fair coin the number of combinations of the number of heads The binomial distribution frequency distributions for the binomial The expected value of a discrete random variable the variance of a discrete random variable 9 Concept Bernoulli Trial two outcomes e g success or failure successive independent trials probability of success is the same in each trial Example flipping a coin multiple times 10 Flipping a Coin Once The random variable k is the number of heads it is variable because k can equal one or zero it is random because the value of k depends on probabilities of occurrence p and 1 p Prob p Heads k 1 Prob 1 p 11 Tails k 0 Flipping a coin once Expected value of the number of heads is the value of k weighted by the probability that value of k occurs E k 1 p 0 1 p p variance of k is the value of k minus its expected value squared weighted by the probability that value of k occurs 12 VAR k 1 p 2 p 0 p 2 1 p VAR k 1 p p 1 p p 1 p p Flipping a coin twice 4 elementary outcomes Prob p Prob p heads heads h h h hk 2 Prob 1 p tails Prob 1 p 13 h h t k 1 t Prob p heads t t h h k 1 Prob 1 p tails t t t t k 0 tails Flipping a Coin Twice Expected number of heads E k 2 p2 1 p 1 p 1 1 p p 0 1p 2 E k 2 p2 p p2 p p2 2p so we might expect the expected value of k in n independent flips is n p Variance in k VAR k 2 2p 2 p2 2 1 2p 2 p 1 p 0 2p 2 1 p 2 14 Continuing with the variance in k VAR k 2 2p 2 p2 2 1 2p 2 p 1 p 02p 2 1 p 2 VAR k 4 1 p 2 p2 2 1 4p 4p2 p 1 p 4p2 1 p 2 adding the first and last terms 8p2 1 p 2 2 1 4p 4p2 p 1 p and expanding this last term 2p 1 p 8p2 1 p 8p3 1 p VAR k 8p2 1 p 2 2p 1 p 8p2 1 p 1 p so VAR k 2p 1 p or twice VAR k for 1 flip 15 So we might expect the variance in n flips to be np 1 p 16 Frequency Distribution for the Number of Heads A fair coin 17 One Flip of the Coin probability 1 2 O heads 18 1 head of heads Two Flips of a Fair Coin probability 1 2 1 4 0 19 1 2 of heads Three Flips of a Fair Coin It is not so hard to see what the value of the number of heads k might be for three flips of a coin zero one two three But one head can occur two ways as can two heads Hence we need to consider the number of ways k can occur I e the combinations of branching probabilities where order does not 20 count Three flips of a coin 8 elementary outcomes p p p H H 1 p H H T p 1 p T T H H T 1 p p T H 1 p T T 3 heads 2 heads 2 heads 1 head 2 heads 1 head 1 head 0 heads Three Flips of a Coin There is only one way of getting three heads or of getting zero heads But there are three ways of getting two heads or getting one head One way of calculating the number of combinations is Cn k n k n k Another way of calculating the number of combinations is Pascal s triangle 22 23 Three Flips of a Coin Probability 3 8 2 8 1 8 0 24 1 2 3 of heads The Probability of Getting k Heads The probability of getting k heads along a given branch in n trials is p k 1 p n k The number of branches with k heads in n trials is given by Cn k So the probability of k heads in n trials is Prob k Cn k pk 1 p n k This is the discrete binomial distribution where k can only take on discrete values of 0 1 k 25 Expected Value of a discrete random variable n E x x i p x i i 0 the expected value of a discrete random variable is the weighted average of the observations where the weight is the frequency of that observation Expected Value of the sum of random variables E x y E x E y 27 Expected Number of Heads After Two Flips Flip One kiI heads Flip Two kjII heads Because of independence p kiI and kjII p kiI p kjII Expected number of heads after two 1 k I k II flips E kiI kjII 1 i j p kiI p kjII i 0 E kiI kjII 1 i 0 j 0 kiI p kiI p kjII 1 j 0 Cont 1 E …
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