# Discrete Random Variable

• Feb 21st 2011, 09:22 PM
dwsmith
Discrete Random Variable
The median annual income for the heads of households in a certain city is $44,000. Four such heads of households are randomly selected for inclusion in an opinion poll. Let X be the number (out of four) who have annual incomes below$44,000.

Find the probability distribution.

$\displaystyle \displaystyle\frac{\binom{4}{x}\binom{4}{4-x}}{\binom{y}{4}}, \ \ \ x=0, \ 1, \ 2, \ 3, \ 4$

I don't know what y is.
• Feb 21st 2011, 10:43 PM
Random Variable
Notice that they said "median" and not "mean." So it's binomial with p = 0.5 and n = 4.
• Feb 21st 2011, 11:42 PM
CaptainBlack
Quote:

Originally Posted by dwsmith
The median annual income for the heads of households in a certain city is $44,000. Four such heads of households are randomly selected for inclusion in an opinion poll. Let X be the number (out of four) who have annual incomes below$44,000.

As Random_Variable points out this is a binomial RV with p=0.5 and n=4 (because $44,000 is the median) Quote: Find the probability distribution.$\displaystyle \displaystyle\frac{\binom{4}{x}\binom{4}{4-x}}{\binom{y}{4}}, \ \ \ x=0, \ 1, \ 2, \ 3, \ 4$I don't know what y is. This last part makes no sense is it supposed to be related to the first part? How? CB • Feb 22nd 2011, 10:27 AM dwsmith Quote: Originally Posted by CaptainBlack As Random_Variable points out this is a binomial RV with p=0.5 and n=4 (because$44,000 is the median)

This last part makes no sense is it supposed to be related to the first part? How?

CB

The probability distribution p(x) is for x = 0, 1, 2, 3, 4.

So my numerator would be 4 choose x times the 4 choose 4-x divided by the sample population which would be population choose 4 but I don't know the population.
• Feb 22nd 2011, 01:28 PM
CaptainBlack
Quote:

Originally Posted by dwsmith
The probability distribution p(x) is for x = 0, 1, 2, 3, 4.

So my numerator would be 4 choose x times the 4 choose 4-x divided by the sample population which would be population choose 4 but I don't know the population.

That is not the mass function for the binomial distribution.

CB
• Feb 22nd 2011, 05:11 PM
dwsmith
Quote:

Originally Posted by CaptainBlack
That is not the mass function for the binomial distribution.

CB

Then I don't know what I am supposed to do.
• Feb 22nd 2011, 06:13 PM
CaptainBlack
Quote:

Originally Posted by dwsmith
Then I don't know what I am supposed to do.

As we have said $\displaystyle$$X$ is a RV with a binomial distribution $\displaystyle B(p=0.5, n=4)$

$\displaystyle pr(X=x)=b(x;p,n)=\dfrac{n!}{x!(n-x)!}p^x(1-p)^{n-x};\ \ \ \ n=4, p=0.5; x=0,1,..4$

CB