Question

How do you find conditional distributions?
We did this chart in class in which we categorized males and females into age groups.
http://puu.sh/b6gBt/2e82e79ffd.png As in this chart.
Then he asked us to find the "conditional distributions". I know that conditional distribution means that you have to take a number of two categories (For example,Females 40-49) but I have no idea which number to divide them into. Do you divide the total number of people into the number of females 40-49? The number of females? The number of people ages 40-49? Thank You!

Answer 1

\[\begin{array}{|c|c|c|c|c|}
\hline&20-29&30-39&40-49&50-59&\bf\text{Total}\\
\hline
\text{Male}&4&7&4&5&\bf20\\
\hline
\text{Female}&6&8&6&0&\bf20\\
\hline
\bf\text{Total}&\bf10&\bf15&\bf10&\bf5&\bf40\\
\hline
\end{array}\]
The 20-29 column is one conditional distribution. For example, the probability that a person randomly picked from the 40 total people falls in the 20-29 range is \(\dfrac{10}{20}=\dfrac{1}{2}\). The probability that such a person falls in the 20-29 range, given that it's a male, is
\[\frac{P(\text{20-29 and male})}{P(\text{male})}=\frac{4/40}{20/40}=\frac{4}{20}=\frac{1}{5}\]
Another conditional distribution is the male row. For example, the probability that a randomly picked person is male, given that he/she is 20-29 years old is
\[\frac{P(\text{male and 20-29})}{P(\text{20-29})}=\frac{4/40}{10/40}=\frac{4}{10}=\frac{2}{5}\]

Answer 2

Oooh ok that makes sense now...so it could be either of the two categories. So basically if the book asks you "find the conditional distributions" I should get each individual number and do those two divisions? So this problem would have... 16 conditional distributions?

Answer 3

8 distributions. Gender given age, and age given gender. 4 ranges times 2 genders gives 8 distributions.

Answer 4

Sorry to keep bothering you but...how is it 8? I mean the one you just did, doesn't that count as 2 conditional distributions? And if you do that for each number, that means that there would be 16....unless the two you just did count as 1?

Answer 5

Sorry, I made a mistake with the counting... My mind is stuck in combinatorics :P

The actual number of conditional distributions is \(\Large6\), not 8 or 16.

A conditional distribution fixes one variable and gives the probability that another variable takes on a certain value. It basically comes down to counting the number of rows and columns.

Here's one cond. dist.:
\[\begin{array}{|c|c|c|c|c|}
\hline&\color{red}{20-29}&30-39&40-49&50-59&\bf\text{Total}\\
\hline
\text{Male}&\color{red}4&7&4&5&\bf20\\
\hline
\text{Female}&\color{red}6&8&6&0&\bf20\\
\hline
\bf\text{Total}&\bf10&\bf15&\bf10&\bf5&\bf40\\
\hline
\end{array}\]
In this case, we've fixed the age to be one category, and we'd examine the distribution of gender among the age range.

Another would be if we fixed the gender and looked at the distribution of ages:
\[\begin{array}{|c|c|c|c|c|}
\hline&20-29&30-39&40-49&50-59&\bf\text{Total}\\
\hline
\text{Male}&4&7&4&5&\bf20\\
\hline
\color{red}{\text{Female}}&\color{red}6&\color{red}8&\color{red}6&\color{red}0&\bf20\\
\hline
\bf\text{Total}&\bf10&\bf15&\bf10&\bf5&\bf40\\
\hline
\end{array}\]
So two rows and four columns gives 6 total conditional distributions.

Answer 6

Oh thank you!! (Sorry for replying so late, OpenStudy was giving me a bit of trouble with logging in >o<) I understand it now :). I got a 100 on the quiz too, thanks so much!!