Class Note for STAT 528 at OSU 54
Class Note for STAT 528 at OSU 54
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Date Created: 02/06/15
Stat 528 Autumn 2008 Elly Kaizar Random Variables Reading Sections 43 44 0 Random variables RVs Discrete and continuous RVs o The mean of a RV The mean of discrete RV The mean of some continuous RVs Transformations Rules for means 0 The variance and standard deviation of a RV Rules for variances Random variables 0 Recall A variable is any characterisitc of an individual A variable can take different values for different individuals77 0 A random variable RV X is a variable that depends on the outcome of a chance experiment or a random phe nomenon A random variable must have a numeric value 0 We can have discrete or continuous RVs 0 Examples Number of heads in three coin ips Height of students selected at random from the stat 528 class ln a random sample of components the number that pass a test Number of particles counted by a Geiger counter in a radiation experiment Discrete random variables o A discrete random variable X takes on a nite or sometimes countable number of values as o The probability distribution of X is a table of probabil ities associated with each value of X valueofX 12 33 33k probability p1 p2 p3 pk This table is also called a probability mass function 0 By the rules of probability we know that 1 2511 1 2 For each 239 0 g pzr g 1 c We can also present the probability distribution of X using a probability histogram a histogram of the values of X versus the probability 0 We calculate the probability of events by summing up the probabilities pzr for the values 36 that make up that event Telephone example Suppose that the length X of an international telephone call to the nearest minute is given by valueolel 2 3 4 probability OQ 05 02 01 Calculate the following 1 13032 2 PX lt 2 3 PX 1 Random walk example A y leaves a restaurant Every minute thereafter the y ran domly moves either 1 meter left 1 with probability 05 or 1 meter right 1 with probability 05 Let the RV X denote the distance the y moves left or right in three minutes relative to his start position What is the probability distribution of X Random walk example c0nt Continuous random variables o A continuous RV X takes values 13 anywhere in an inter val of values This interval could be unbounded oo 0 Example Consider the direction of a spinner What is the probability that the spinner lands between 900 and 1800 0 Probability distributions for continuous RVs are de scribed by the probability density curve 1 The density curve always has nonnegative height 2 The area under the density curve is one Compare with the probability distribution for discrete RVs Calculating probabilities o Probabilities are given by the area under the curve eg Pa lt X lt b is o For any one value 13 of X because there is no area at one point But the height under the probability density curve need not be zero The uniform distribution o A continuous RV X has a uniform distribution if it has probability density curve 0 a and b are the parameters of the uniform distribution 0 We say that X has a Ua 9 distribution The normal distribution 0 See the previous notes 0 Note When answering questions using the normal probabil ity distribution we should be careful to phrase our answers carefully eg Plta g X g 9 rather than a g X g b 10 SRS example An opinion poll asks a SR8 of 1500 adults do you happen to jog77 Suppose that in fact 15 of adults would answer yes to this question However the proportion 1 of the sample who answer yes in this sample will vary in repeated sampling We will show later in this Class that we can suppose that I is normally distributed with mean u 015 and standard deviation 0 00092 Find the probability that either less than 14 or over 16 of the polled adults claim to jog 11 SRS example c0nt 12 The mean or expected value of a discrete RV 0 Suppose we have a discrete RV with probability distribution valueofX ZQ 2 3 33k probability p1 p2 p3 pk 0 Then the mean or expected value of X is k MX Zion 21 0 lntuition Supposep pfor alli1k 113 k HX 2213323913239 13 Discrete Mean Example 0 Ex Remember that the length X of an international tele phone call to the nearest minute is given by valueolel 2 3 4 probability 02 05 02 01 What is the mean length of an international telephone call 14 The mean of a transformation of a discrete RV 0 Ex Now suppose that an international telephone call costs 8 cents per minute With a 5 cent connection charge What is the average or mean charge for an international call 0 Let Y be some transformation of X Then the mean or expected value of Y is k MY 211239 1 21 0 EX cont Let Y be the charge for an international tele phone call With Y 8X 5 The distribution of X and Y is value of X 1 2 3 4 value of Y probability 02 05 02 01 0 Thus My 15 The mean of a continuous RV o Harder to calculate need calculusl FYI My f 13 f 93 f probability density function 0 The normal and uniform distributions are both symmetric ln these cases the mean is equal to the median Thus The mean of a NW 0 RV is u The mean of a Ua 9 RV is a b2 draw the picture for each case 16 Rules for means 0 Let X and Y be discrete or continuous RVs Then MXY HX My 0 Let a and b be xed numbers Then HabX a bMX note this is a linear transformation 0 Example Look again at the telephone example Where we calculate My Where Y 8X 5 17 The variance and standard deviation of a RV 0 Let X be a RV o The distance X is away from its mean uX is X MX o The square of this distance is X MX o The variance of a RV X is the mean of this squared dis tance 2 i 0X WX MX o If X is a discrete RV then k air 235239 MX2 1 21 If X is continuous need to use calculus again 0 The standard deviation of X 0X is the square root of the variance of X 18 The variance and stdev of certain continuous RVs 0 You should know that The variance of a NW 0 RV is 02 and the stdev is a The variance of a Ua 9 RV is b 0212 and so the standard deviation is b a 19 Rules for variances 0 Let X and Y be discrete or continuous RVs and let a and b be xed numbers Then 2 i 2 2 0abX b 0X39 0 IF X and Y are independent then 2 i 2 2 UXY70XUy 0 IF X and Y are not independent then you need to worry about the correlation between X and Y for more details about what is meant by the correlation between two random variables see Section 22 20 Calcium example The level of calcium in the blood of healthy young adults follows a normal distribution with mean u 10 milligrams per deciliter and standard deviation 0 05 a What proportion of healthy young adults have a calcium level that lies between 95 and 10 milligrams per deciliter hint draw the picture 21 Calcium example cont Two young adults are drawn at random from the population of all healthy young adults Let X denote the calcium level of the rst adult and Y denote the calcium level of the second adult b What is the mean and standard deviation of the difference W X Y 22