stats_ch6_notes

stats_Chapter 6.pdf

• COMMA MEANS AND IN THIS CONTEXT

Joint probility distro p(x,y)

• 2 discrete rand vars [[stats_ch4_notes Discrete Random Variablrs#Discrete Random Variablrs dpqr#Defitions Random Vs Discrete Random Vs Continous Random]]
• gives p(x,y) for all combos of x & y, rem prob must equate to 1 (sum of all poss)

What I should be able to do given table below”:

• AND OR GIVEN MARGINAL
• p(y=1,x=2?) = sum

$Y=y$ \ $X=x$	1	2	3	4
0	0	.10	.20	.10
1	.03	.07	.10	.05
2	.05	.10	.05	0
3	0	.10	.05	0

Joint Prob

Joint Marginal prob aka $p_1(x)$

• slide 9/10

Varience of $y_1$ can be denoted like so:

Joint density func example, An easy double integral;\

Suppose the joint density function for two continuous random variables, $X$ and $Y$, is given by

$$f(x,y)=\{\begin{array}{ll}cx& \text{if }0\le x\le 1;0\le y\le 1\\ 0& \text{elsewhere}\end{array}$$

Determine the value of the constant $c$.

Solution:

1. We note the inequalities btwn 0&1
2. Using our given function, we can generate a graph with it, where
   1. only cx generates some output, and all other places generate a 0 (via area, so generate some triangle $f(x,y)=cx$ with 2 lines)
3. Now that we have the shape, we can integrate the triangle, where we know that the area must equal 1 (since it rep total prob space)
4. Hence, we can just integrate the double integral easily.. (recall triangles)

• Note we must generate the correct picture! A graph of $f(x,y)$ traces a three-dimensional, wedge-shaped figure over the unit square ($0\le x\le 1$ and $0\le y\le 1$) in the ($x$, $y$)-plane, as shown in Figure 6.1. The value of $c$ is chosen so that $f(x,y)$ satisfies the property ${\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x,y)dxdy=1$ Performing this integration yields

$$\begin{array}{c}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }f(x,y)dxdy={\int }_0^1{\int }_0^{1}cxdxdy\\ =c{\int }_0^1{\int }_0^{1}xdxdy=c{\int }_0^1{\left[\frac{x^2}{2}\right]}_0^{1}dy\\ =c{\int }_0^1\frac{1}{2}dy=\left(\frac{c}{2}\right)[y{]}_0^1=\frac{c}{2}\end{array}$$

Covarience Defition (p30)

Cumlative distro func:

See ex 6.15 (in exam!)

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$$\begin{array}{c}{F}_w(w)=..\\ f_w(w)=\frac{d{F}_w(w)}{dw}=..\end{array}$$

Then we plot normally (p func) and density (d func). We denote this by using the

plot()