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###Sample spaces and probability

###Random variables

###Expectation

###Law of the Unconscious Statistician

###Answers

Sample spaces and probability

Random variables

Expectation

Law of the Unconscious Statistician

Answers

for all numbers $x$ and $y$. The LOTUS now involves two-dimensional real integrals, frequently expressed as Riemann or Lebesgue integrals. The concept of expectation needs to be generalized slightly, though, because the ordered pair $(X,Y)$ is never just a number. Instead, let $h$ be a function of two real variables. Provided that $h$ is measurable and the chance of $h(X,Y)$ being undefined is zero (which permits, among other things, the possibility of analyzing functions like $h(x,y)=x/y$ which are not defined when $Y=0$), $h(X,Y)$ is another random variable. This is meant in exactly the same sense $Z=XY$ is a random variable: it's a way of combining the numbers $X(\omega)$ and $Y(\omega)$ into a third number $h(X(\omega), Y(\omega))$ for each $\omega$ (throwing away any $\omega$ for which this combination is undefined, provided the set of such $\omega$ has zero probability). Now

for all numbers $x$ and $y$. The LOTUS now involves two-dimensional real integrals, frequently expressed as Riemann or Lebesgue integrals. The concept of expectation needs to be generalized slightly, though, because the ordered pair $(X,Y)$ is never just a number. Instead, let $h$ be a function of two real variables. Provided that $h$ is measurable and the chance of $h(X,Y)$ being undefined is zero (which permits, among other things, the possibility of analyzing functions like $h(x,y)=x/y$ which are not defined when $y=0$), $h(X,Y)$ is another random variable. This is meant in exactly the same sense $Z=XY$ is a random variable: it's a way of combining the numbers $X(\omega)$ and $Y(\omega)$ into a third number $h(X(\omega), Y(\omega))$ for each $\omega$ (throwing away any $\omega$ for which this combination is undefined, provided the set of such $\omega$ has zero probability). Now

for all numbers $x$ and $y$. The LOTUS now involves two-dimensional real integrals, frequently expressed as Riemann or Lebesgue integrals. The concept of expectation needs to be generalized slightly, though, because the ordered pair $(X,Y)$ is never just a number. Instead, let $h$ be a function of two real variables. Provided it is measurable, $h(X,Y)$ is another random variable, in exactly the same sense $Z=XY$ is a random variable: it's a way of combining the numbers $X(\omega)$ and $Y(\omega)$ into a third number $h(X(\omega), Y(\omega))$ for each $\omega$. Now

or (for discrete distributions) as double sums. The function $\frac{\partial^2}{\partial x\partial y}F(x,y)$ (if it exists) is called the bivariate density of $(X,Y)$.

for all numbers $x$ and $y$. The LOTUS now involves two-dimensional real integrals, frequently expressed as Riemann or Lebesgue integrals. The concept of expectation needs to be generalized slightly, though, because the ordered pair $(X,Y)$ is never just a number. Instead, let $h$ be a function of two real variables. Provided that $h$ is measurable and the chance of $h(X,Y)$ being undefined is zero (which permits, among other things, the possibility of analyzing functions like $h(x,y)=x/y$ which are not defined when $Y=0$), $h(X,Y)$ is another random variable. This is meant in exactly the same sense $Z=XY$ is a random variable: it's a way of combining the numbers $X(\omega)$ and $Y(\omega)$ into a third number $h(X(\omega), Y(\omega))$ for each $\omega$ (throwing away any $\omega$ for which this combination is undefined, provided the set of such $\omega$ has zero probability). Now

(with a double sum appearing instead of a double integral for discrete distributions). The function $\frac{\partial^2}{\partial x\partial y}F(x,y)$ (if it exists) is called the bivariate density of $(X,Y)$.