# Probability density function problem

1. Jan 27, 2014

### Nikitin

1. The problem statement, all variables and given/known data
Let the probability density function$f(x) = (3/4) \cdot (1-x^2)$ if x is between -1 and 1, and let $f(x)=0$ otherwise.

What is the probability of $P(X \leq 0.8 | X>0.5)$?
2. Relevant equations

3. The attempt at a solution

I assume I have to rewrite the p.d.f. into a joint probability density function so I can use bayes' rule?

And what does the large "$X$" mean in probabilities? Often my texts uses large X-es instead of small. Is the large X for the actual event, while the small x is just a function variable?

2. Jan 27, 2014

### LCKurtz

$X$ denotes the random variable whose density function is $f(x)$. So the probability that $X$ falls between $a$ and $b$ is$$P(a\le X \le b) = \int_a^b f(x)~dx$$You should be able to use that formula along with the definition of conditional probability to solve your problem.

3. Jan 27, 2014

### Nikitin

Could you put me on the right track? I'm not sure how to use conditional probability on this, as I don't know the probability for the intersection between "$X \leq 0.8$" and "$X>0.5)$".

4. Jan 27, 2014

### LCKurtz

What values of $X$ satisfy both inequalities?

5. Jan 27, 2014

### Ray Vickson

Draw a number line for X. Sketch the two regions X > 0.5 and X ≤ 0.8. What part of the line is the region common to both these larger regions? Using the given density, how would you compute the probability of that common region?

I urge you to struggle with this if necessary; asking for too much help on such problems hinders your learning and will not be good for you in the long run at exam time. If you cannot get it in 2 minutes, don't give up. If you need two hours, take two hours.

6. Jan 27, 2014

### Nikitin

oh god how foolish of me. Thank you! I understand it all now.

I got it now. I was too confused to see what the intersection of the two events actually was (it was really stupid of me to not get it immediately). thanks for the advice!