Download Random Variables Handout

Discrete
Continuous
Support
Countable set of values
Uncountable set of values
Probabilities
pmf: P (X = x) = p(x)
P (X = x) = 0 6= f (x) for all x
Rb
P (a ≤ X ≤ b) = a f (x)dx = F (b) − F (a)
P
Expected Value µX = E[X] = x xp(x)
P
E[g(X)] = x g(x)p(x)
R∞
µX = E[X] = −∞ xf (x) dx
R∞
E[g(X)] = −∞ g(x)f (x) dx
Independence
fXY (x, y) = fX (x)fY (y)
pXY (x, y) = pX (x)pY (y)
Table 1: Differences between Discrete and Continous Random Variables
Probability Mass Function p(x)
Probability Density Function f (x)
Discrete Random Variables
Continuous Random Variables
p(x) = P (X = x)
f (x) 6= P (X = x) = 0
p(x) ≥ 0
f (x) ≥ 0
p(x) ≤ 1
P
x p(x) = 1
P
F (x0 ) = x≤x0 p(x)
f (x) can be greater than one!
R∞
f (x) dx = 1
−∞
R x0
F (x0 ) = −∞
f (x) dx
Table 2: Properties of probability mass function (pmf) versus probability density function.
1
2
Set of all values the RV can take
F (x0 ) = P (X ≤ x0 )
2
σX
= V ar(X) = E (X − E[X])2
Support
CDF
σXY = Cov(X, Y ) = E [(X − E[X]) (Y − E[Y ])]
X, Y indep. ⇒ Cov(X, Y ) = 0 but Cov(X, Y ) = 0 ; X, Y indep.
Definition of Std. Dev.
Covariance
Cov. and Independence
ρXY = Corr(X, Y ) = σXY /(σX σY )
E[g(X)] 6= g (E[X])
E[a + bX] = a + bE[X] where a, b are constants and X is a RV
Definition of Correlation
Expectations of Functions
Linear Functions
Table 3: Essential facts that hold for all random variables, continuous or discrete
V ar(aX + bY ) = a2 V ar(X) + b2 V ar(Y ) + 2abCov(X, Y ) for any RVs X, Y and constants a, b
V ar(X1 + . . . + Xk ) = V ar(X1 ) + . . . V ar(Xk ) if X1 , . . . , Xk are independent RVs
E[X1 + . . . + Xk ] = E[X1 ] + . . . E[Xk ] where X1 , . . . , Xk are any RVs
V ar(a + bX) = b2 V ar(X) where a, b are constants and X is a RV
Cov(X, Y ) = E[XY ] − E[X]E[Y ]
Shortcut for Covariance
Functions and Independence X, Y indep. ⇒ g(X), h(Y ) indep. where g, h are any functions
Shortcut for Variance
V ar(X) = E[X 2 ] − (E[X])2
p
2
σX = σX
Definition of Variance
X : S → R (RV is a fixed function from sample space to reals)
Definition of R.V.
3
sXY
n
p
s2X
1 X
(xi − x̄)2
n − 1 i=1
σX =
2
σX
=
p
σx2
N
1 X
(xi − µX )2
N i=1
Population viewed as list of objects
N
1 X
µX =
xi
N i=1
Population Parameter
ρXY = σXY /(σX σY )
n
N
1 X
1 X
=
(xi − x̄)(yi − ȳ) σXY =
(xi − µX )(yi − µY )
n − 1 i=1
N i=1
sX =
s2X =
Sample from a population
n
1X
x̄ =
xi
n i=1
Correlation rXY = sXY /(sX sY )
Covariance
Std. Dev.
Variance
Mean
Setup
Sample Statistic
µX =
−∞
R∞
p
σx2
xf (x) dx
ρXY = σXY /(σX σY )
σXY = E [(X − µX ) (Y − µY )]
σX =
2
σX
= E (X − E[X])2
Continuous
x
Population viewed as a RV
X
Discrete µX =
xp(x)
Population Parameter