How do you calculate marginal, joint, and conditional probabilities from a two-way table?

1 Answer
May 18, 2015

If you are given a pmf = pXY(x,y)

and you would like to find the marginal pY(y)

we would use the formula py(y)=ip(xi,y)

in other words you would sum over all of x at the point y

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So if we look at this table and want to find the marginal pY(3)

we go:

pY(3)=P(Y=3)
=P(Y=3,X=3)+P(Y=3,X=4)
=0.1+0.2
=0.3

Now to look at the formula for the conditional probability

we can look at the formula for x given y which is a conditional probability.

pXY(xy)=P(X=xiY=yj)=P(X=xi,Y=yj)P(Y=yj)

=pXY(xi,yj)pY(yi)

now to use an example, we will look back at our table.

let us look for the conditional probability of:

pXY(34)=0.10.4=0.25

Thus, the probability that X=3 given that Y=4 is 0.25