1. ## Continuous random variables

Hi, I have this math assignment on continuous random variables and would greatly appreciate some help. This is the question that I am having trouble with:

Sara loves shoes and buys many pairs, but gets tired of them quickly. She throughs them away by the pair only into a dumpster, which is emptied every 3 months. Let X be the number of shoes in the dumpster at any given time and let Y be a continuous random variable which is known to be a good approximation of X.

(a.) whats the numerical value of Pr[X=9] ?
(b.) Find an expression in terms of Y for each of the following:

(i) Pr[X=40] (ii) Pr[X<40] (iii) Pr[X>36]

(c.) If Pr[Y>21] = 1-Pr[X<k], what is the value of k?

Thanks!!

2. Originally Posted by rebeccajm
Hi, I have this math assignment on continuous random variables and would greatly appreciate some help. This is the question that I am having trouble with:

Sara loves shoes and buys many pairs, but gets tired of them quickly. She throughs them away by the pair only into a dumpster, which is emptied every 3 months. Let X be the number of shoes in the dumpster at any given time and let Y be a continuous random variable which is known to be a good approximation of X.

(a.) whats the numerical value of Pr[X=9] ?
As the shoes are thrown out in pairs there will always be an even
number of shoes in the dumpster so Pr[X=9]=0.

(b.) Find an expression in terms of Y for each of the following:

(i) Pr[X=40] (ii) Pr[X<40] (iii) Pr[X>36]

(c.) If Pr[Y>21] = 1-Pr[X<k], what is the value of k?

Thanks!!
To do these last parts we need to know what you know
about the description of probabilities of continuous random variables.

Are you familiar with the distribution function (aka the cumulative distribution)?
Or the probability density function?

RonL
the

3. I am slightly familiar with the cumulative distribution function. I have done a question which involved making a chart and graphing both the probability distribution fumction as well as the cummulative distribution function. As far as how to use it in this particular question, I am pretty lost. Thanks for your help!

4. Originally Posted by rebeccajm

(b.) Find an expression in terms of Y for each of the following:

(i) Pr[X=40] (ii) Pr[X<40] (iii) Pr[X>36]
Let $\displaystyle Q(y)=Pr(Y<y)$ be the cumulative probability function for
the random variable $\displaystyle Y$ Then as $\displaystyle Y$ is a good approximation for
$\displaystyle X$ we would usualy use the approximation:

$\displaystyle Pr(X=40) \approx Pr(39.5<Y<40.5)=Q(40.5)-Q(39.5)$

However there is a complication here in that the probability that $\displaystyle X$
take an odd value is $\displaystyle 0$, so to maintain proper nomalisation of
the probabilities our approximation should probably be:

$\displaystyle Pr(X=40) \approx Pr(39<Y<41)=Q(41)-Q(39)$

Continuing in this manner we have:

$\displaystyle Pr(X<40)\approx Pr(Y<39)=Q(39)$,

and:

$\displaystyle Pr(X>36)\approx Pr(Y>37)=$$\displaystyle 1-Pr(Y \le 37)=1-Pr(Y < 37)=1-Q(37)$

(we are using the result that $\displaystyle Pr(Y<37)=Pr(Y \le 37)$ for a continuous RV)

RonL

5. Originally Posted by rebeccajm
(c.) If Pr[Y>21] = 1-Pr[X<k], what is the value of k?
The way we have been working, if $\displaystyle k$ is even:

$\displaystyle Pr(X<k) \approx Pr(Y<(k-1))$,

so:

$\displaystyle Pr(Y>21) = 1-Pr(Y<21) \approx 1-Pr(X<22)$

Hence $\displaystyle k=22$.

RonL