Download Joint Probability Calculations in R

Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Joint Probability Calculations in R
M. George Akritas
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Marginal and Conditional PMFs through R
pXY=c(.388 , .485, .090 , .009 , .010, .008 ,.003, .005, .002) #
the vector of joint probabilities from the rat-drug example
pXYm =matrix(pXY,3,3) # the matrix of joint probabilities
px=rowSums(pXYm); py=colSums(pXYm) # the marginal PMFs
diag(1/px) %*% pXYm # the matrix pY |X =x (y )
x,y
pXYm[1,]/px[1] # the first row of pY |X =x (y )
x,y
pXYm %*% diag(1/py) # the matrix pX |Y =y (x)
x,y
pXYm[,1]/py[1] # the first column of pX |Y =y (x)
M. George Akritas
x,y
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
The joint PMF of independent RVs in R
For X ∼ Bin(2, 0.5), Y ∼ Bin(3, 0.5) independent, do:
pXYm=dbinom(0:2,2,0.5) %*% t(dbinom(0:3,3,0.5))
Alternatively:
P=expand.grid(px=dbinom(0:2,2,0.5),py=dbinom(0:3,3,0.5));
P$pxy=P$px*P$py; pXYm =matrix(P$pxy,3,4)
Check: P$pxy same as pXY=as.vector(pXYm)
Also check:
px=rowSums(pXYm) ; py=colSums(pXYm) # same as
dbinom(0:2,2,0.5) and dbinom(0:3,3,0.5), respectively.
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
The PMF and CDF of X + Y with R
I
Let X ∼ Bin(3, 0.3) and Y ∼ Bin(4, 0.6), X , Y independent.
P=expand.grid(px=dbinom(0:3,3,.3),py=dbinom(0:4,4,.6));
P$pxy=P$px*P$py
G=expand.grid(X=0:3,Y=0:4); P$X=G$X; P$Y=G$Y; attach(P)
sum(pxy[which(X+Y==4)]) # pmf of X + Y at x = 4
sum(pxy[which(X+Y<=4)]) # cdf of X + Y at x = 4
sum(pxy[which(3<X+Y & X+Y<=5)]) # P(3 < X + Y ≤ 5)
I
Try: v=seq(0,20,2) ; v[5] ; which(v == 8) ; which(2<= v &
v<= 8); sum(v[2:5]) ; sum(v[which(2<= v & v<= 8)])
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Another Conditional PMF with R
I
With X and Y as in the previous slide, find the conditional
PMF of X given that X + Y = 6.
Solution: Given X + Y = 6, the possible values of X (i.e. the
conditional sample space) are 3 and 2.
X[which(X+Y==6)] # gives the conditional sample space
px[which(X+Y==6)]*py[which(X+Y==6)]/sum(pxy[which(X+Y==6)])
# gives the conditional pmf,
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Proposition
Each of the following statements implies, and is implied by, the
independence of X and Y .
1. pY |X (y |x) = pY (y ).
2. pY |X (y |x) does not depend on x, i.e. is the same for all
possible values of X .
3. pX |Y (x|y ) = pX (x).
4. pX |Y (x|y ) does not depend on y , i.e. is the same for all
possible values of Y .
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Verification of the proposition with R
Use the joint PMF of X ∼ Bin(2, 0.5) and Y ∼ Bin(3, 0.5) , X , Y
independent.
pXYm[1,]/rowSums(pXYm)[1] # gives P(Y = y |X = 0), y = 0 : 3
Check that this is the same as
dbinom(0:3,3,.5) # why?
Next verify that the following are the same:
pXYm[2,]/rowSums(pXYm)[2] # gives P(Y = y |X = 1), y = 0 : 3
pXYm[3,]/rowSums(pXYm)[3] # gives P(Y = y |X = 2), y = 0 : 3
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Mean value of a function in R
pxy=c(0.10, 0.08, 0.06, 0.04, 0.20, 0.14, 0.02, 0.06, 0.30)
pxym=matrix(pxy,3,3)
px=rowSums(pxym) ; py=colSums(pxym)
P=expand.grid(px=px,py=py) ; P$pxy=pxy
G=expand.grid(x=0:2,y=0:2) ; P$x=G$x; P$y=G$y
attach(P) ; sum((x+y)*pxy)
sum(x*pxy) ; sum(y*pxy)
sum((x+y)**2*pxy)
sum(pmin(x,y)*pxy) ; sum(pmin(x,y)**2*pxy)
Also, E (e X +Y ) = sum(exp(x+y)*pxy) etc.
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Proposition
Let X1 , . . . , Xn be a simple random sample from a population
having variance σ 2 . Then
n
E S 2 = σ 2 , where S 2 =
2
1 X
Xi − X .
n−1
i=1
The next slide gives a numerical verification of this proposition.
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Proof by Simulation
I
Random number generation can be used to provide numerical
evidence in support of, or in contradiction to, certain
probabilistic/statistical claims.
I
Evidence for the above proposition can be obtained by
generating a large number of samples (of any given size and
from any distribution), compute the sample variance for each
sample, and average the sample variances.
I
We will use 10,000 samples of size 5 from the standard
normal distribution. The R commands are:
> m=matrix(rnorm(50000),ncol=10000)
> mean(apply(m,2,var))
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Try the following:
integrate(dnorm, -Inf, Inf) ; integrate(dnorm, -Inf, Inf)$value
integrate(dnorm, -Inf, 1.1) # same as pnorm(1.1)
Define your own function and integrate it:
f = function(x) {1/((x+1)*sqrt(x))}
integrate(f, lower = 0, upper = 5)
Can also do derivatives (symbolic and algorithmic): Type ?deriv
M. George Akritas
Joint Probability Calculations in R
Outline
Joint Probability Calculations in R
Mean Value of Functions of Random Variables
Extra Topic: Integration in R
Double Integrals
The following commands compute
R 0.5 R y
0
−0.5 tan(x
+ y )dxdy :
g1=function(y){integrate(function(x) tan(x+y), -0.5, y)$value}
# the inside integral
g2=function(y) {sapply(y,g1)} # note g2(y)=g1(y) for all y
integrate(g2,0,.5)
What is missing is the ability to define
g=function(x,y){tan(x+y)*(x¡y)} # this you can do
and then use a command like
integrate(g, over=(-0.5, 0.5)x(0, 0.5))
M. George Akritas
Joint Probability Calculations in R