# 1.3 Joint and conditional cdfs and pdfs

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This module introduces joint and conditional cdfs and pdfs

## Cumulative distribution functions

We define the joint cdf to be

$F(x, y)=((X\le x)\land (Y\le y))$
and conditional cdf to be
$F(|(x, y))=(Y\le y, X\le x)$
Hence we get the following rules:
• Conditional probability (cdf) :
$F(|(x, y))=(Y\le y, X\le x)=\frac{F(x, y)}{{F}_{Y}(y)}$
• Bayes Rule (cdf) :
$F(|(x, y))=\frac{F(|(y, x))F(x)}{F(y)}$
• Total probability (cdf) :
$F(x)$ F x
which follows because the event $Y$ itself forms a partition of the sample space.
Conditional cdf's have similar properties to standard cdf's, i.e. ${F}_{X|Y}(|(()))$ y 0 ${F}_{X|Y}(|())$ y 1

## Probability density functions

We define joint and conditional pdfs in terms of corresponding cdfs. The joint pad is defined to be

$f(x, y)=\frac{\partial^{2}F(x, y)}{\partial x\partial y}$
and the conditional pdf is defined to be
$f(|(x, y))=\frac{\partial^{1}\frac{d F(|(x, Y=y))}{d }}}{\partial x}$
where $\frac{d F(|(x, Y=y))}{d }}=(Y=y, X\le x)$ Note that $\frac{d F(|(x, Y=y))}{d }}$ is different from the conditional cdf $F(|(x, Y=y))$ , previously defined, but there is a slight problem. The event, $Y=y$ , has zero probability for continuous random variables, hence probability conditional on $Y=y$ is not directly defined and $\frac{d F(|(x, Y=y))}{d }}$ cannot be found by direct application of event-based probability. However all is OK if we consider it as a limitingcase:
$\frac{d F(|(x, Y=y))}{d }}=\lim_{\delta (y)\to 0}(y< Y\le y+\delta (y), X\le x)=\lim_{\delta (y)\to 0}\frac{F(x, y+\delta (y))-F(x, y)}{{F}_{Y}(y+\delta (y))-{F}_{Y}(y)}=\frac{\frac{\partial^{1}F(x, y)}{\partial y}}{{f}_{Y}(y)}$
Joint and conditional pdfs have similar properties andinterpretation to ordinary pdfs: $f(x, y)> 0$ $\int \int f(x, y)\,d x\,d y=1$ $f(|(x, y))> 0$ $\int f(|(x, y))\,d x=1$
From now on interpret $\int$ as ${\int }_{-\infty }^{\infty }$ unless otherwise stated.
For pdfs we get the following rules:
• Conditional pdf:
$f(|(x, y))=\frac{f(x, y)}{f(y)}$
• Bayes Rule (pdf):
$f(|(x, y))=\frac{f(|(y, x))f(x)}{f(y)}$
• Total Probability (pdf):
$\int f(|(y, x))f(x)\,d x=\int f(y, x)\,d x=f(y)\int f(|(x, y))\,d x=f(y)$
The final result is often referred to as the Marginalisation Integral and $f(y)$ as the Marginal Probability .

what is the stm
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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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