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Probability Distributions Guide

This document provides an overview of several probability distributions and concepts: 1) It defines the Poisson, Bernoulli, negative binomial, uniform, exponential, gamma, and normal distributions. It gives the probability mass or density functions and expected values and variances for each. 2) It covers concepts like indicator variables, functions of random variables, Markov's and Chebyshev's inequalities, joint and marginal distributions, and order statistics. 3) It states properties of independent random variables and expectations including that expected values of independent variables multiply and variances add.

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54 views2 pages

Probability Distributions Guide

This document provides an overview of several probability distributions and concepts: 1) It defines the Poisson, Bernoulli, negative binomial, uniform, exponential, gamma, and normal distributions. It gives the probability mass or density functions and expected values and variances for each. 2) It covers concepts like indicator variables, functions of random variables, Markov's and Chebyshev's inequalities, joint and marginal distributions, and order statistics. 3) It states properties of independent random variables and expectations including that expected values of independent variables multiply and variances add.

Uploaded by

guanyuc
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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P(AIBC P

Bayesian theorem:P(A/B)
ANB) ASPA)
=

Discrete:

poisson distruction:discrete random variable. f(Kix) Pr(X=k)


=

**
=

E(X) Var(x) 5
=
=

Bernoulli distribution:f(k:p)=pR(H-p)H E(X) p =

Var (x) P(rp) p9


=
=

puzzionaminliCan(PRCPIU E(x) npVar(x):npg xp (59)


=

=11 -

ps-.pdiscrete. Ex Var x =
Negative Biomial:P(X n)
(951) pK(1-p(n-k E(x) *
=

=
=

5a Xca
Continuous;
Uniform distribution. f(x):a F(x) =

Exponential distribution:
(xy
xx
x70
pdf f(X x) cdf:F(Xix)(1 -2*x
=
=

xcp
E(x) *
=
Var(x) =
Gamma distribution: f(x) 2a) x8
=
-

e-x*for x20 U(a) fota--+d+


=

(9,N) (1,5) is
exporential
Normal distributionsfy, 62)
f(x) xpe-1x-URKS.
=

N10.1) standard normal:


f(x) ye x
= -

·
for continuous distribution:p(Xx) 0 for X =
=

· Indicator Variable 1) A 25! if Aoccussee not occur

·
Functions of a random variable:Y=gix) Ycannot have more values than X

Var (x) E((X-E(X(R)


=

(789x(x).
=

(x- (uxdX =
5px(x). (x (2x)" Variatbx) PVar(x)
-

E(X) 3x px(x) f8xfx(x)dX Var (x) E(XY E(X)


= -
=

= -

Markov's inequality:P(XF+) Y=t.13xIt).ECY(=0.P(Y= 0) ++. PLYt) t.P(X=t)


=
=
=
1E(X)
=

·
Chelysher's inequality:P11X-(0x (It) G
Jointdistributions:
marginal conditional marginal:Fx(x) P(X<X)
=

independent if Fx, y(x,y) Fx(x)Fy (y)


=

IPIAN Bi) conditional:P(A)


=

if independent E(X-Y) E(X) E(Y) =


ZP (A/Bi)P(Bi) -

E(X.Y)
Sx(yx.xx,y(x,y)dy.dX SxSyx-x.8x(x)fy(y)dydx 1)(xxfx(x)dX). ((yy.fy(y)dy) E(X).E(Y)
=
=

=
=

also, for function g.bglx)hlY) EGg(Xh(Y) Eg(X)C.ECh/Y1S =

order statistics: Fxcy(x) =


1 -

11 -

F(X))
"

COUNX,Y) E( (X(X). (Y (Y)) = F(XY) ENCELY) cour(X,Y) CoIXY


-
-
= =

Var (x+Y) Var (x) +Var(Y) 2001X,Y)


= +

Var (x1 +x2 +


... -
Xn) EVar(xi) +2Z
= Ov(Xi, Xj)
jj
XAY IS COU(X.Y) 0
=

Var(x+Y) Var(x) Var(Y)


=
+

Conditional Expectation:
E(X(Y x) 5,xfx(y(x(Y)dxTower law:E(Y) ECELY (X)
=
=

Law of total variance:Var(Y) Elar(Y(x)S VarlE(Y(x() =


+

E (VarLY(x) VarIY) =

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