How Can i simulate realizations ?

Question

if i have Two Discrete Random variables X and Y have the joint PMF:

Pxy=c(x^2+y^2)

where x belong to {1,2,4} and y belong to {1,3)

how can I simulated variables by computing the relative frequencies and comparing it to the joint PMF

Answer 1

Prob(x,y) = c(x^2+y^2)

Prob(1,1) = 2c

Prob(1,3)= 10c

Prob(2,1) = 5c

Prob(2,3)= 13c

Prob(4,1)= 17c

Prob(4,3)= 25c

BUT, these probabilities MUST total 100%

2c+10c+5c+13c+17c+25c = 1 = 100%

72c = 1

c = 1/72

So Prob(x,y) = (1/72)(x^2 + y^2)

and

Prob(1,1) = 2c = 1/36 = 2/72 = 2+7/9% --> call it event A

Prob(1,3)= 10c = 5/36 = 10/72 = 13 + 8/9%--> call it event B

Prob(2,1) = 5c = 5/72 = 6.94% ---> call it event C

Prob(2,3)= 13c = 13/72 = 18.05%---> call it event D

Prob(4,1)= 17c = 17/72 =23.61%---> call it event E

Prob(4,3)= 25c = 34.72%---> call it event F

You can now generate a random number between ZER0 and 72

to simulate these SIX (6) events A,B,C,D,E,F having these

prescribed probabilities, per the table below

random # event realized/occurred

------------------------------

[0,2) A

[2,12) B

[12,17) C

[17,30) D

[30,47) E

[47,72) F

I have uploaded a spreadsheet that does just this.

It is in the RESOURCES section under this link.