Many phenomena in nature have approximately Normal distributions.

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1 NORMAL DISTRIBUTION The Normal r.v. plays an important role in probability and statistics. Many phenomena in nature have approximately Normal distributions. has a Normal distribution with parameters and, denoted by N(, ) if f ( x) e - x - ; 0

2 NORMAL DISTRIBUTION: bell shaped and symmetric around the mean/median/mode The parameter μ is the center (or mean) of the distribution σ is the spread (or standard deviation) of the distribution.

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6 The Standard Normal Let N(, ), Z=(- μ )/σ is the Standard Normal rv with mean 0 and variance 6

7 For instance, given N(, ) P( a a b) P( b a ) P( Z b ) Example. A discount store sells DVD. Historical sales records indicate that the weekly demand,, for DVD is normally distributed with mean μ=00 and standard deviation σ= Calculate the probability that is greater than Solution: standardize x to get z, from the table get the corresponding probability (direct problem) 0 00 P( 0) P( Z ) P( Z ) 0 The area for Z> is 587 (Use the table of the Standard Normal) ( 7

8 From the table we get P(Z)=843 so P(Z>)=-P(Z)=-843=587 8

9 How many DVDs should be stocked at the beginning of a week so that there is only a 5% chance that the store will run short of DVD during the week? Solution: from the table get the standardized value z, from z get the corresponding x (inverse problem) From the table we get P(Z>.645)=05 so z=.645 and x=.645(0)+00=6.45. z.645 x 00 0 The store should stock at least 7 DVDs. 9

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16 Bivariate r.v. Bivariate distributions For example: roll two dices. To each outcome we have a couple of values (=x;y=y) so to have a Bivariate random variable. Y /36 /36 /36 /36 /36 /36 /6 /36 /36 /36 /36 /36 /36 /6 3 /36 /36 /36 /36 /36 /36 /6 4 /36 /36 /36 /36 /36 /36 /6 5 /36 /36 /36 /36 /36 /36 /6 6 /36 /36 /36 /36 /36 /36 /6 /6 /6 /6 /6 /6 /6 Joint probability distribution P(=x, Y=y) Marginal distribution of Marginal distribution Y 6

17 Bivariate r.v. Two random variables and Y are independent if: P( =x and Y=y)=P(=x)P(Y=y) for each couples of values x, y Y For example: roll two dices, and Y P(=odd and Y=even)=9/36=5 P(=odd)=P(, 3, 5)=3/6=5 P(Y=even)=P(, 4, 6)=3/6=5 3 P(=odd)P(Y=even)=5x5=5 4 and Y are independent

18 Bivariate r.v. CONDITIONAL DISTRIBUTIONS We can compute the conditional distribution of given that we have observed Y = y. P( x For discrete random variables we have P( Y y), y) P( y) Which are the conditional distributions on party identification P( Y=woman) and P( Y=man)? Y Gender Political party Democrat Independent Republican Total Women Men Total

19 Bivariate r.v. CONDITIONAL DISTRIBUTIONS Y Gender Political party Democrat Independent Republican Total Women 50/50=33 70/50=47 30/50= Men 40/70=4 50/70=9 80/70=47 9

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21 COVARIANCE COVARIANCE If and Y have finite variance, their covariance is defined as Cov(,Y) = E[( - E( ))(Y - E(Y))] Covariance is a measure of association. A positive covariance indicates that when is above (respectively below) its mean, also Y will tend to be above its own mean. Viceversa, a negative covariance indicates that when is above (respectively below) its mean, also Y will tend to be below its own mean. If and Y are independent than Cov(,Y)=0, the viceversa is not true

22 COVARIANCE CORRELATION If and Y have finite variance, their correlation is defined as r(, Y) COV (, Y) VAR( ) VAR( Y) Correlation is a measure of association, but it is normalized so as to take values in the range -, only. If r(,y)= indicates a perfect positive correlation, if r (,Y)=- indicates a perfect negative correlation A perfect correlation exists when =a+by (or Y=a+b) If and Y are independent than r (,Y)=0, the viceversa is not true

23 COVARIANCE Covariance of two rv and Y: For two discrete rv: i j i j σ Y =Cov(,Y)=E[(-E())(Y-E(Y))]=E(Y)-E()E(Y) ( x E( ))( y E( Y)) P( x, Y y ) i j ( x y ) P( x, Y y ) E( ) E( Y) i j i i i i For example, trend of a price of a bond () and a price of a stock (Y) in a month (+=rise; -=fall) Y Y E( Y) E( ) E( Y) Y E( Y) E( ) E( Y) ( )( 4) 63 3

24 Bivariate r.v. Conditional expectation of given Y=y j : where E( Y y j) xip( xi Y y j) P( x is the conditional probability i x Y y j P( xi, Y y ) P( Y y ) j For example, trend of a price of a bond () and a price of a stock (Y) in a month (+=rise; -=fall) Y E( Y ) 75 E Y ) E( Y ) 67 E Y ) j ) ( 43 ( 65 4

25 Bivariate r.v. Conditional variance of given Y=y j : x E( Y y ) P( x Y y ) V( Y y j) i j i j x For example, trend of a price of a bond () and a price of a stock (Y) in a month (+=rise; -=fall) Y E( Y ) V( Y ) ( 75) ( 75)

26 Bivariate r.v. Linear combination of random variables Property of expectation operator: Y=a +a + +a k k E(Y)=a E( )+a E( )+ +a k E( k ) For example, trend of a price of a bond () and a price of a stock (Y) in a month (+=rise; -=fall) E( ) W=-Y E(W)=E()-E(Y)=4-4=6 6

27 Bivariate r.v. 7 Linear combination of random variables Y=a +a + +a k k Property of variance If,,, k are not independent: k k k k k k k k k k k k a a a a V a V a V a a a a a a a V a V a V a Y V cov( ) cov( ) ( ) ( ) ( ) cov( ) cov( ) cov( ) ( ) ( ) ( ) ( For example, trend of a price of a bond () and a price of a stock (Y) in a month (+=rise; -=fall) V() =96 V(Y)=98 Cov(Y)=63 W=-Y V(W)=4V()+V(Y)+()(-)Cov(Y)= =.3 Note: -3 W 3

28 Bivariate r.v. Linear combination of random variables Property of variance If,,, k are independent: Y=a +a + +a k k V( Y) a V( ) av( ) akv( k) For example, two dice rolled, and E( )=E( )=3.5 V( )=V( )=.9 Y= E(Y)=E( )-E( )=7-3.5=3.5 V(Y)=4V( )+V( )=.68+.9=4.6 8

29 Bivariate r.v. Linear combination of Normal random variables Given Y=a +a + +a k k where,,, k are independent normal rvs with then i ~ N( i, i ) Y ~ N(a a ak k,a a ak k ) 9