Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet; it is ot perfectly predictable. A discrete radom variable ca take oly a fiite umber of values, that ca be couted by usig the positive itegers. A cotiuous radom variable ca take ay real value (ot just whole umbers) i a iterval o the real umber lie. Slide 2.
2.2 The Probability Distributio of a Radom Variable Whe the values of a discrete radom variable are listed with their chaces of occurrig, the resultig table of outcomes is called a probability fuctio or a probability desity fuctio. For a discrete radom variable X the value of the probability desity fuctio f(x) is the probability that the radom variable X takes the value x, f(x)=p(x=x). Therefore, 0 f(x) ad, if X takes values x,.., x, the f( x) + f( x2) + L+ f( x ) =. For the cotiuous radom variable Y the probability desity fuctio f(y) ca be represeted by a equatio, which ca be described graphically by a curve. For cotiuous radom variables the area uder the probability desity fuctio correspods to probability. Slide 2.2
2.3 Expected Values Ivolvig a Sigle Radom Variable 2.3. The Rules of Summatio. If X takes values x,..., x the their sum is 2. If a is a costat, the i 2 L i= x = x + x + + x 3. If a is a costat the i= a = a ax = a x i i= i= i Slide 2.3
4. If X ad Y are two variables, the 5. If X ad Y are two variables, the ( x + y ) = x + y i i i i i= i= i= ( ax + by ) = a x + b y i i i i i= i= i= 6. The arithmetic mea (average) of values of X is Also, x xi i= x+ x2 + L + x. = = i= ( x x) = 0 i Slide 2.4
7. We ofte use a abbreviated form of the summatio otatio. For example, if f(x) is a fuctio of the values of X, i= f( x ) = f( x ) + f( x ) + L+ f( x ) i 2 = f ( x ) ("Sum over all values of the idex i") = i x i f ( x) ("Sum over all possible values of X") 8. Several summatio sigs ca be used i oe expressio. Suppose the variable Y takes values ad X takes m values, ad let f(x,y)=x+y. The the double summatio of this fuctio is m m f ( x, y ) = ( x + y j ) i j i i= j= i= j= Slide 2.5
To evaluate such expressios work from the iermost sum outward. First set i= ad sum over all values of j, ad so o. That is, To illustrate, let m = 2 ad = 3. The ( i, j) = ( i, ) + ( i, 2) + ( i, 3) 2 3 2 f x y f x y f x y f x y i= j= i= (, ) (, 2) (, 3) (, ) + (, ) + (, ) = f x y + f x y + f x y + f x y f x y f x y 2 2 2 2 3 The order of summatio does ot matter, so m m f ( x, y ) = f( x, y ) i j i j i= j= j= i= Slide 2.6
2.3.2 The Mea of a Radom Variable The expected value of a radom variable X is the average value of the radom variable i a ifiite umber of repetitios of the experimet (repeated samples); it is deoted E[X]. If X is a discrete radom variable which ca take the values x, x2,,x with probability desity values f(x ), f(x2),, f(x), the expected value of X is EX [ ] = xf( x) + xf( x) + L+ xf( x) = = 2 2 i= x xf( x) i xf ( x) i (2.3.) Slide 2.7
2.3.3 Expectatio of a Fuctio of a Radom Variable If X is a discrete radom variable ad g(x) is a fuctio of it, the EgX [ ( )] = gx ( ) f( x) (2.3.2a) x However, EgX [ ( )] gex [ ( )] i geeral. Slide 2.8
If X is a discrete radom variable ad g(x) = g (X) + g 2 (X), where g (X) ad g 2 (X) are fuctios of X, the EgX [ ( )] = [ g( x) + g( x)] f( x) x 2 (2.3.2b) = g ( x) f( x) + g ( x) f( x) x 2 x = Eg [ ( x)] + Eg [ ( x)] 2 The expected value of a sum of fuctios of radom variables, or the expected value of a sum of radom variables, is always the sum of the expected values. If c is a costat, Ec [] = c (2.3.3a) If c is a costat ad X is a radom variable, the Slide 2.9
EcX [ ] = cex [ ] (2.3.3b) If a ad c are costats the Ea [ + cx] = a+ cex [ ] (2.3.3c) 2.3.4 The Variace of a Radom Variable [ ] 2 var( X ) EgX [ ( )] E X EX ( ) EX [ ] [ EX ( )] 2 2 2 =σ = = = (2.3.4) Let a ad c be costats, ad let Z = a + cx. The Z is a radom variable ad its variace is 2 2 var( a+ cx) = E[( a+ cx) E( a+ cx)] = c var( X) (2.3.5) Slide 2.0
2.4 Usig Joit Probability Desity Fuctios 2.4. Margial Probability Desity Fuctios If X ad Y are two discrete radom variables the f( x) = f( x, y) for each value X ca take y f( y) = f( x, y) for each value Y ca take x (2.4.) 2.4.2 Coditioal Probability Desity Fuctios f ( xy, ) f( x y) = P[ X = x Y = y] = (2.4.2) f ( y) Slide 2.
2.4.3 Idepedet Radom Variables If X ad Y are idepedet radom variables, the f ( xy, ) = f( xf ) ( y) (2.4.3) for each ad every pair of values x ad y. The coverse is also true. If X,, X are statistically idepedet the joit probability desity fuctio ca be factored ad writte as f 2 2 2 ( x, x, L, x ) = f ( x ) f ( x ) K f ( x ) (2.4.4) Slide 2.2
If X ad Y are idepedet radom variables, the the coditioal probability desity fuctio of X give that Y=y is f( x, y) f( x) f( y) f ( x y) = f( x) f( y) = f( y) = (2.4.5) for each ad every pair of values x ad y. The coverse is also true. 2.5 The Expected Value of a Fuctio of Several Radom Variables: Covariace ad Correlatio If X ad Y are radom variables, the their covariace is cov( X, Y) = E[( X E[ X])( Y E[ Y])] (2.5.) Slide 2.3
If X ad Y are discrete radom variables, f(x,y) is their joit probability desity fuctio, ad g(x,y) is a fuctio of them, the EgXY [ (, )] = gxy (, ) f( xy, ) (2.5.2) x y If X ad Y are discrete radom variables ad f(x,y) is their joit probability desity fuctio, the cov( X, Y) = E[( X E[ X])( Y E[ Y])] = [ x E( X)][ y E( Y)] f( x, y) x y (2.5.3) Slide 2.4
If X ad Y are radom variables the their correlatio is cov( XY, ) ρ = (2.5.4) var( X ) var( Y ) 2.5. The Mea of a Weighted Sum of Radom Variables EaX [ + by] = ae( X) + bey ( ) (2.5.5) If X ad Y are radom variables, the E[ X + Y] = E[ X] + E[ Y] (2.5.6) Slide 2.5
2.5.2 The Variace of a Weighted Sum of Radom Variables If X, Y, ad Z are radom variables ad a, b, ad c are costats, the 2 2 2 [ ax + by + cz ] = a [ X ] + b [ Y ] + c [ Z ] + 2abcov [ X, Y ] + 2accov [ X, Z ] + 2bccov [ Y, Z ] var var var var (2.5.7) If X, Y, ad Z are idepedet, or ucorrelated, radom variables, the the covariace terms are zero ad: [ ax by cz ] a 2 [ X ] b 2 [ Y ] c 2 [ Z ] var + + = var + var + var (2.5.8) Slide 2.6
If X, Y, ad Z are idepedet, or ucorrelated, radom variables, ad if a = b = c =, the [ X Y Z] [ X] [ Y] [ Z] var + + = var + var + var (2.5.9) 2.6 The Normal Distributio 2 ( x β) f( x) = exp, < x< 2 2σ 2 2 πσ (2.6.) Z = X β σ ~ N(0,) Slide 2.7
If X ~ N(β, 2 σ ) ad a is a costat, the X a a PX [ a] P β β P Z β = = σ σ σ (2.6.2) If X ~ N(β, 2 σ ) ad a ad b are costats, the a X b a b Pa [ X b] P β β β P β Z β = = σ σ σ σ σ (2.6.3) If X ~ N(, 2 ), X ~ N(, 2 ), X ~ N(, 2 ) β σ 2 β2 σ2 3 β3 σ 3 ad c, c, c3 are costats, the ( ) ( ) Z = cx + ~,var cx + 2 2 cx 3 3 N E Z Z (2.6.4) 2 Slide 2.8