# Thread: Continuous probability - conditional probability

1. ## Continuous probability - conditional probability

Suppose that $Y$ has density function

$f(y)=\left\{\begin{array}{cc}ky(1-y),&0\leq y\leq 1\\0, & \mbox{elsewhere}\end{array}\right.$

a. Find the value of $k$that makes $f(y)$ a probabilty density function.
No problem, found $k=6$
b. Find $P(.4 \leq Y \leq 1) = \int^{1}_{.4}6y(1-y)dy=.648$

the part I am having trouble with is;
d. Find $P(Y \leq .4 | Y \leq .8)$
not too sure if $P(Y \leq .4 | Y \leq .8)=\frac{P(.4 \leq Y \leq .8)}{P(Y \leq .8)}$ or how to calculate this conditional probability...

2. Originally Posted by Robb
Suppose that $Y$ has density function

$f(y)=\left\{\begin{array}{cc}ky(1-y),&0\leq y\leq 1\\0, & \mbox{elsewhere}\end{array}\right.$

a. Find the value of $k$that makes $f(y)$ a probabilty density function.
No problem, found $k=6$
b. Find $P(.4 \leq Y \leq 1) = \int^{1}_{.4}6y(1-y)dy=.648$

the part I am having trouble with is;
d. Find $P(Y \leq .4 | Y \leq .8)$
not too sure if $P(Y \leq .4 | Y \leq .8)=\frac{P(.4 \leq Y \leq .8)}{P(Y \leq .8)}$ Mr F says: This should be ${\color{red}P(Y \leq .4 | Y \leq .8)=\frac{P(Y \leq .4 \, \text{ and } \, Y \leq .8)}{P(Y \leq .8)}}$.

or how to calculate this conditional probability...
Note that $P(Y \leq .4 \, \text{ and } \, Y \leq .8) = P(Y \leq .4)$.

Therefore $P(Y \leq .4 \, | \, Y \leq .8) = \frac{P(Y \leq .4)}{P(Y \leq .8)}$ and you should be able to calculate those two things.