On one of the feet? 1 2. On red? 1 4. Within 1 of the vertical black line at the top?( 1 to 1 2

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1 Continuous Random Variable If I spin a spinner, what is the probability the pointer lands... On one of the feet? 1 2. On red? 1 4. Within 1 of the vertical black line at the top?( 1 to 1 2 )? 360 = Within one millionth of a degree of that line? 360,000,000 With arrow pointing exactly vertical???!??!?!?0 I guess Easy to give probs it falls in range of values (proport. to length). But probs of exact value don t make sense, or maybe are 0. I want you to notice that your intuition totally works here. The fact that the spinner chooses from an infinite continuum of points makes talking about probability tricky, but it is perfectly clear to everyone if you divide the circle up into four (or any number) equal sized pieces that each piece is equally likely to get pointed at, just like the doors in Let s Make a Deal. There are infinitely many points on the circle, and surely they are all equally likely, so if each one has any positive probability of occurring the probabilities would not add up to 1 (in fact they would add up to infinity!). So we cannot find the probability of one point, but we can find the probability of an interval made up of infinitely many points!!! Continuous Random Variable - The Brilliant Trick Continuous random variables like the time it takes you to run a mile, suffer from the same problem. It makes sense to ask what the probability of taking between 8 and 9 minutes, but not the probability of exactly 9 minutes. We need a trick to talk about the probability of a range of values without talking about individual values. Towards that trick, consider a histogram of women s heights with bins of... 4 inches 1 inch.25 inches In the limit it approaches some continuous function!#)"!#('"!#("!#'"!#(&"!#(%"!#($"!#&"!#("!#%"!#$"!#!$" '*" (%" ) )'" )*" +%" +)" %'" %)" * *(" *$" *+" *%" **" *&" *," *'" *)" & &(" &$" &+" &%" &*" &&" &," &'" &)",! 1

2 Remember continuous random variable can take on any real number as a value. Since there are infinitely many real numbers between every two real numbers, any interval has infinitely many points in it, so it cannot make sense to talk about the probability of one particular real number occurring. There are infinitely many points on the circle, and surely they are all equally likely, so if each one has any positive probability of occurring the probabilities would not add up to 1 (in fact they would add up to infinity!). So we cannot find the probability of one point, but we can find the probability of an interval made up of infnitely many points!!! As you break your data up into finer and finer bins, they shape tends to converge to a nice smooth shape. I am cheating a little. To make this work you have to scale the vertical axis so that the total area of the histogram is 1 at each stage of the process. Continuous Distributions We represent a continuous random variable by a continuous function f(x) called its distribution, which we can think of like a histogram. It must be never negative and the total area under the curve must be 1. The probability X falls in a range of values (interval) is area under the curve above the interval. For example the area above represents probability of a height being between 60 and 66 inches P (60 < X < 66)!#!!$" (*" $ $&" $'" $$" $+" $," $)" + +%" +&" +'" +(" +$" ++" +," +)" +*",,%",&",(",$" Continuous Distributions and Area The area under the graph of a function f(x) between x = a and x = b goes by a name you ve seen before. Integral! b a f(x)dx Important point: The value of the function f(x) at x means nothing (it is not P (X = x), that is 0), only its integral tells you something meaningful. 2

3 The reason area under the curve works so well is that, as we observed with Venn diagrams, the rules of probability work like the rules of area. The Normal Distribution Many random variables in nature follow a recognizable bell-shaped distribution. There are deep mathematical reasons for this, as we will see, and they point to one particular function f(x) as the standard bell-shaped distribution. It is called the normal distribution (or the Gaussian distribution. That guy Gauss again!). There is one normal distribution for each µ and σ, with mean µ and standard deviation σ. Any random variable we meet which is symmetric and unimodal we will model by the normal distribution with the same mean and standard deviation. We model things by the normal distribution based only on their shape. We will see situations that for theoretical reasons we know we can model by the normal distribution. The Normal Distribution-Definition The normal distribution with mean µ and standard deviation σ is given by the following distribution function: f(x) = 1 σ 1 2π e 2( x µ σ ) 2 Notice it depends only on x s z-score! The complicated coefficient in front is just to make the total integral f(x)dx come out to 1. This might be the most beautiful calculation in all of undergraduate mathematics! You remember e = the result of a dollar spending a year at 100% interest compounded continuously, and π = , the ratio of the circumference of a circle to its diameter. But what are they doing here, in a stats class? Is nowhere safe from these irrational monstrosities? Some examples of normal distributions for various values of µ and σ. All come from one by shifting and stretching in x-direction. 3

4 The Normal Distribution-Definition f(x) = 1 σ 2π e 1 2( x µ σ ) 2!#!!$" (*" $ The mean plus and minus the standard deviation aways occur where the curve switches from curving downward to curving upwards: inflection points! The mean is at a critical point. The probability that a normal random variable will fall between a and b is b a $&" $'" $$" f(x)dx We don t know how to integrate that, in fact can be proven there is no formula whose derivative is function f(x) above. No prob., integrate numerically, i.e. approximate area as closely as we like. A big calculation, but Excel loves that. $+" $," $)" + +%" +&" +'" +(" +$" ++" +," +)" +*",,%",&",(",$",,",)",*" ) The dirty secret of calculus is that almost no functions you need to integrate in real life can be integrated by the techniques you learn in calculus. The few days you spent on numerical integration were the only techniques you learn that are useful most of the time. Normdist Compute probabilities for normal distribution with mean µ and s. d. σ in Excel using P (X x) = normdist(x, µ, σ, 1) x is the number you are comparing it to µ is the mean of the distribution σ is the standard deviation of the distribution 1 is a meaningless ritual, an invocation of the Excel muse From this you can get a couple of formulas P (X b) = normdist(b, µ, σ, 1) P (a X) = 1 normdist(a, µ, σ, 1) P (a X b) = normdist(b, µ, σ, 1) normdist(a, µ, σ, 1) What about <? Continuous dist: < same prob as. The difference between P (X < 4) and P (X 4) is P (X = 4), which for continuous distributions is 0. 4

5 Normdist Ex: Women s heights have a bell-shaped distribution with a mean of 65.5 and a standard deviation of 2.5. What is the probability a woman s height is between 61 and 64 inches. Between 70 and 73 inches? P (61<X <64) = normdist(64, 65.5, 2.5, 1) normdist(61, 65.5, 2.5, 1) = 23.8% Good habit: 64 is less than a s. d. below the mean. 61 is between 1 and 2 s. d. s below the mean, so between 61 and 64 is going to be definitely less than half of area but not a tiny fraction so 23.8% seems about right.!#!!$" ()#*$" (+#$" $!#&$" $%#*$" $&#$" $'#&$" $(#*$" $$#$" $,#&$" $*#*$" $)#$" $+#&$",,!#*$",%#$",&#&$",'#*$",(#$",$#&$",,",,#*$",*#$",)#&$",+#*$" *!#$" *%#&$" *&" *&#*$" *'#$" *(#&$" *$" *$#*$" *,#$" **#&$" *)" *)#*$" *+#$" -./0" %(1" ''1" ''1" %(1" P (70<X <73) = normdist(73, 65.5, 2.5, 1) normdist(70, 65.5, 2.5, 1) = 3.46%!#!!$" 73 more than 3 s.d. s above mean, 70 is almost 2, so near top 2.5%. ()#*$" (+#$" $!#&$" $%#*$" $&#$" $'#&$" $(#*$" $$#$" $,#&$" $*#*$" $)#$" $+#&$",,!#*$",%#$",&#&$",'#*$",(#$",$#&$",,",,#*$",*#$",)#&$",+#*$" *!#$" *%#&$" *&" *&#*$" *'#$" *(#&$" *$" *$#*$" eyeballing the value gets pretty easy once you have done it a few times, and then it is worth it. 5

6 Norminv Occasionally you want to go the other way. For example, if women s heights have a bell-shaped distribution with a mean of 65.5 and a standard deviation of 2.5, what is the 80th percentile of height? That s going the other way because we are asking for b such that P (X < b) =.8. Find this with Norminv b = norminv(p, µ, σ) 67.6 = norminv(.8, 65.5, 2.5) b is the value (height) P is the percentile µ is the mean of the distribution σ is the standard deviation of the distribution P =.8 is the percentile µ = 65.5 is the mean σ = 2.5 is the standard deviation a = 67.6 is the 80th percentile of women s heights. Empirical Rule Redux Remember the Empirical Rule for symmetric unimodal distributions: About 95% of data falls within two standard deviations of the mean. For a normal distribution, we can make that more precise. 95% of the data in a normal distribution fall within 1.96 standard deviations of the mean. You will need to remember that fact, we will use it a lot. In general if C is a probability, zc is the name for the number of s.d.s up and down you have to go to encompass C of the data, so z.95 = The formula is z C = NORMINV((1 C)/2, 0, 1). but you will not be held responsible for that. Lecture on Continuous and Normal Distributions Key Points After watching this lecture you should be able to Recognize when a variable can be modeled by the normal distribution (when it is symmetric and unimodal and you know its mean and s.d.) Roughly estimate normal probabilities from the Empirical Rule. Calculate in Excel probabilities of a range of values and percentiles for normal distributions After processing you should be able to Relate calculations to z-scores 6