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# Probability for Life Sciences Students MATH 3C

UCLA

GPA 3.55

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This 4 page Study Guide was uploaded by Kaylin Wehner on Friday September 4, 2015. The Study Guide belongs to MATH 3C at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 54 views. For similar materials see /class/177840/math-3c-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.

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Date Created: 09/04/15

Math 3G2 Final Study Guide Updated 61007 Written by Akemi Kashiwada Final Wednesday June 13 2007 3 6pm Disclaimer This is just a guide to help you study for the nal The material in this study guide will not necessarily be on the midterm and vice versa This is just what I would knowstudy ifl were a student in Math 30 There may be some typosmistakes so do not just use this to study ie be sure to look at old homeworks notes book etc Material Covered Sections 124126 material covered in Homework 59 to review material covered before the midterm refer to the Midterm Study Guide Section 124 Discrete Random Variables and Discrete Distribuions Before the midterm we discussed up to expected value of a discrete random variable and relative frequencies Recall that we can think of the expected value of some random variable X as a measure of centrality eg center of mass balance The variance of X measures how spread out the distribution is from the mean and can be calculated by varX EX 7 m2 EX2 7 EX2 where M EX Know how to derive these formulasl From variance we can get the standard deviation sd a m Some properties of the expected value and variance If a b are constants and X Y random variables then 0 EaXb aEXb o varaX b a2varX o EX Y EX EY If we want to investigate the relationship between two different discrete random variables then we need to nd their joint distribution Typically we are given a chart from which we need to use relative frequencies to get the joint probability distribution Given Chart Joint Prob Dist where s abcd Each entry of thejoint distribution is denoted PX I Y y and can be interpreted as the probability that X 1 AN y From a joint distribution we can recover the distributions of X and Y called marginal distributions by adding the desired column or row PXz ZPXIYy PYy ZPXIYy Math 30 Discussions 2A X 2B 2 We also have conditional probabilities P X I Y y P X I Y lt w w H w and we say that X7 Y are independent if PX LY y PX zPY y for all possible 17y If X7 Y are independent then we get the properties 0 EXY EXEY o varX Y varX varY Example 1 1 You roll a die twice Let X be the random variable that gives the sum of the two numbers Find the expected value and variance of X 2 Suppose you conduct a survey to determine there is a relationship between gender and favorite movie genre et 0 likes action 0 male X 7 1 39 l 39k d i 1 iffemale if 2 es rama likes comedy You summarize your tally in the following table Find the joint probability distribution What is the probability that a randomly chosen person likes drama What is the probability that a randomly chosen male likes action 5 Let XY be independent random variables with given probability mass functions What is the probability that X is even and Y is odd Find EXYEX Y7 varX Y k PX k PY k 1 02 03 2 06 02 5 02 05 Binomial Distribution Suppose you run an experiment that only two outcomes and the probability of success is p You repeat this experiment n times each called a Bernoulli trial and each trial is independent of the rest Then if X counts the number of successes in the n Bernoulli trials then we say X is binomially distributed and RXMlt d em If X is binomially distributed then we can calcuate the mean and variance using EX np7 varX np1 7 p The binomial distribution can be applied when solving a sampling problem with replacement If we have N of object A and M of object B and we are choosing n objects with replacement then the probability that we get kof objectAis k nik n N N PXkltkgt ltNMgt lt17NMgt Math 30 Discussions 2A X 2B 3 Otherwise if we do not allow replacement then I will PX k This is called the hypergeometric distribution Example 2 1 You toss a coin 100 times What is the probability that you get heads 20 times Find the mean and variance 2 A candy bowl contains 8 peppermints and 12 butterscotch You choose 5 candies What s the probability that you get at least two peppermints a if you replace the candy after each pick and b if you keep each candy you choose The last topic from this section is cumulative distribution function Given a probability mass function for X we de ne the cumulative distribution function to be Fz PX S Conversely if we are given the distribution function of a discrete random variable then the probability mass function can be found using PXzPX z7PXltzFzi limiFyl Example 3 Given the following probability mass function nd the correpsonding distribution function Check that your distribution function is the same as the probability mass function Section 125 Continuous Distributions For continuous distributions X can take on any real value which makes it impossible to look at a probability mass function as in the discrete casel lnstead we have a probability density function such that 2 0 an Lfzdz 1 Using this we can calculate the distrubtion function Fz by x Fz 700 Conversely if we are given the distribution function F we can get the density function f by taking the derivative of F if the derivative of F does not exist at a point y then we set Fy 0 The formulas for mean and variance are now EXzfzdz varXLziu2fzdzLz2fxdzilt1zfzdzgt2l Most likely you will need to use integration by parts to solve these integrals and l hospital7s rule to evaluate theml Math 30 Discussions 2A X 2B 4 Example 4 1 Let 7 for z gt 1 Find c so that f is a density function then nd the corresponding distribution function 2 Let Se gx for z gt 0 Find the mean and standard deviation Normal Distribution An important continuous distribution is the normal distribution If X is normally distributed with mean u and standard deviation 0 then its density function is 1 02 mew20 r When u 0 and a 17 then we call this the standard normal and denote its density function goz and distribution function 2 for the rest of this document we will let Z be the standard normally distributed random variable All normal distributions have the following properties 0 is symmetric about u o the maximum of occurs at u o the in ection points of occur at u i a o the probability that X is within h standard deviations of the mean is k FOX 7 m 2w l 068 2 095 3 099 As usual7 we also know that ff l and Fz ff but we do not need to evaluate the integral to nd lnstead7 we use the table given on page 919 Note that the table is only for the standard normal distribution and to nd albc you look in the row labelled ab and the column labelled c To use the table with a general normal distribution X7 you need to rst convert it to Z using PX zPltXT gtltIgtltgtl 039 039 Example 5 1 Exercises 22 25 on page 885 2 Assume that the scores on a test are normally distributed with average 65 and standard deviation 15 What is the probability that a students score is above 889 Find the highest score so 10 of the class got below that score What percentage of students got between 52 and 78 Section 1262 The Central Limit Theorem If a set of independent random variables X1 l Xn have the same distribution then we say that they are independent and identically distributed iidl For example7 if we roll a die n times the Xi could be the outcome of the ith triall Theorem 1 The Central Limit Theorem Suppose X1 l l Xn are d with mean EXZ u and variance varXZ a2 lt 00 If Sn 21Xi then ZSTLHOO

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