Class Note for STAT 528 at OSU 31
Class Note for STAT 528 at OSU 31
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Date Created: 02/06/15
Stat 528 Autumn 2008 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 o A random variable RV X is a numerical variable that depends on the outcome of a chance experiment a random phenomenon 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 o By the rules of probability we know that 1 2511 1 2 For each 239 0 3 pi 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 Examples 0 Telephone calls Suppose that the length X of an inter national telephone call to the nearest minute is given by valueofX 1 2 3 4 probability 02 05 02 01 Calculate the following 1 13032 2 PXlt2 3 13031 0 Random walk A fly leaves a restaurant Every rninute thereafter the fly randomly 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 fly rnoves left or right in three minutes relative to his start position What is the probability distribution of X Continuous random variables o A continuous RV X takes values 13 anywhere in an inter val of values 0 Example X is the direction of a spinner X can take on any value in the range 00 to 3600 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 o Probabilities are given by the area under the curve 0 For any one value 13 of X as the area beneath a single point is O 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 0 Compute probabilities by nding areas The normal distribution 0 See the previous notes 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 of the sample 1 who answer yes will vary in repeated sampling We will later nd 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 adults in the sample jog 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 2 36 pi 21 0 EX Remember that the length X of an international tele phone call to the nearest minute is given by valueofX l 2 3 4 probability 02 05 02 01 What is the mean length of an international telephone call 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 10 The mean of a continuous RV o Harder to calculate need calculusl o The normal and uniform distributions are both symmetric In 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 11 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 bux 0 Example Look again at the telephone example Where we calculate My Where Y 8X 5 12 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 uX MX239 o If X is a discrete RV then k air 235239 MX2 13239 21 0 The standard deviation of X 0X is the square root of the variance of X 13 The variance and stdev of certain continuous RVs 0 You should know that The variance of a N u 02 RV is 02 and so its standard deviation is a The variance of a Ua 9 RV is b 0212 and so its standard deviation is b a 14 Rules for variances 0 Let X and Y be discrete or continuous RVs and let a and b be xed numbers Then 03mm 520x 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 pages 3017305 15 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 o What proportion of healthy young adults have a calcium level that lies between 95 and 10 milligrams per deciliter hint draw the picture 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 0 What is the mean and standard deviation of the difference WX Y 16
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