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,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Discrete and Continuous Random Variables,I can find the standard deviation of discrete random variables.I can find the probability of a continuous random variable.,6.1b,h.w:pg pg 354:14,18,19,23,25,The Variance of a Discrete Random Variable,Recall:,Variance,and,standard deviation,are,measures of spread.,If X is a discrete random variable with mean,then the,variance of X,is,The,standard deviation,is the square root of the variance.,Example:Selling of Aircraft,Gain Communication sells aircraft communications units to both the military and the civilian markets.Next years sales depend on market conditions that can not be predicted exactly.,Gains follows the modern practice of using,probability estimates of sales.,The military estimates the sales as follows:,Units sold:,1000 3000 5000 10,000,Probability:,0.1 0.3 0.4 0.2,Take,X,to be the,number of military units sold.,Compute,x,:,x,=(1000)(0.1)+(3000)(0.3)+(5000)(0.4)+(10000)(.2),=100+900+2000+2000,=,5000 units,Calculate the variance of X:,x,2,=(x,i,-,x,),2,p,i,=(1000 5000),2,(0.10)+finish,=7,800,000,Standard Deviation,x,=sqrt 7,800,000,=2792.8,The,standard deviation,is the measure of,how variable,the number of units sold is.,To find the,variance,with,calculator:,In notes for your info.Try it if you want on your own.,Recall:If we use a table or a calculator to select digits 0 and 1,the result is a discrete random variable which we can“count.,What is the probability of 0.3 X 0.7?,Infinite possible values!,Now we will assign values as,areas under a density curve.,Continuous Random Variables,A,continuous random variable,X,takes all values in,a given interval,of numbers.,The probability distribution of a continuous random variable is shown by a,density curve,.,The,probability that X is between an interval,of numbers is the,area under the density curve between the interval endpoints.,The probability that a,continuous,random variable,X,is,exactly equal to a number is zero,Example:Uniform Distribution,(of random digits between 0 and 1),Note:P(X 0.5 or X 0.8),=0.7,We can add non-overlapping parts.,Normal Distributions as Probability Distributions,Recall,N(,),is the shorthand notation for the,normal distribution,having mean and standard deviation.,If X has the N(,)then the,standardized variable,Z=(X )/,is a standard normal random variable having the distribution,N(0,1),.,Example:Drugs In School,An opinion poll asks a SRS of 1500 of U.S.adults what they think is the most serious problem facing our schools.,Suppose 30%would say“drugs.,The population parameter p,is approximately N(0.3,.0118).,What is the probability that the poll differs from the truth about the population by,more than 2 percentage points,?More than one way to do this.,=P(p 0.32)Theshaded region,=P(p 0.32),Confirm the z-scores and the area of the shaded region.,Standardize the values.,P(,p,0.28)=P(z (0.28 0.30)/0.0118),=P(z 0.32)=0.0455 also why?,P(,p,0.32),=0.0455+0.0455,=0.0910,Conclusion:,The probability that the sample will miss the truth by more than 2 percentage points is,0.0910.,Or,use the complement to get“middle or“unshaded region:,1 P(-1.69 z 2),=P(X=3)+P(X=4)+P(X+5),=0.20,20%of households have more cars than the garage can hold.,
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