Non-Zero Discrete Distributions

[squenshl]

Homework Statement

Suppose we have a discrete random variable whose values $X = x$ can include the value $0$. Some examples are: $X\sim \text{Binomial}(n,p)$ with $x = 0,1,2,\ldots,n$ and $X\sim \text{Poisson}(\lambda)$ with $x = 0,1,2,3,\ldots$ Sometimes we can only observe these variables when $X = x \neq 0$, i.e. when $x$ is not zero. These distributions are known as the non-zero (adjustments) to the original distributions. For the examples discussed above, they are the Non-Zero Binomial and Non-Zero Poisson respectively. Show that the variable $Y = x$ for $x \neq 0$ has:

1. Probability function $\text{Pr}(Y=y)=c\text{Pr}(X = x)$ for $x \neq 0$ and that $c = \frac{1}{1-\text{Pr}(X=0)}$.
2. $E(Y) = cE(X)$.
3. $\text{Var}(Y) = c\text{Var}(X)+c(1-c)\left((E(X)\right)^2$.
4. With respect to the two non-zero examples above, discuss when this adjustment can be ignored, i.e. it has little impact on the what is actually observed.

Homework Equations

$E(Y) = \sum_{i=1}^{n} y_i \text{Pr}(Y=y)$.
$\text{Var}(Y) = E(Y^2)-E(Y)^2$.

The Attempt at a Solution

I'm not actually sure what they are trying to for 1. 2. and 3. Do we have to apply the actual density functions of the 2 respective distributions above or what?

 

[andrewkirk]

There are a couple of ways you could look at this. One is as a conditional distribution: replace
$$E(\textrm{some expression involving }Y),\ Var(\textrm{some expression involving }Y),\ Pr(\textrm{some expression involving }Y)$$
by
$$E(\textrm{some expression involving }X|X\neq 0),\ Var(\textrm{some expression involving }X|X\neq 0),\ Pr(\textrm{some expression involving }X|X\neq 0)$$
respectively.

The other is to define a new probability space. If the original probability space is $(\{0,1,...\},\Sigma, P)$ then define a new space $(\{1,2,...\},\Sigma', P')$ such that, for $A\subseteq \mathbb{N}-\{0\}$, $P'(A)=\frac{P(A)}{1-P(\{0\})}$ and $\Sigma'=\{S-\{0\}\ |\ S\in\Sigma\}$. Then we can define $Y$ as the restriction of $X$ to $\{1,2,...\}$.

It's probably easier to go with the first one.

 

[squenshl]

There are a couple of ways you could look at this. One is as a conditional distribution: replace
$$E(\textrm{some expression involving }Y),\ Var(\textrm{some expression involving }Y),\ Pr(\textrm{some expression involving }Y)$$
by
$$E(\textrm{some expression involving }X|X\neq 0),\ Var(\textrm{some expression involving }X|X\neq 0),\ Pr(\textrm{some expression involving }X|X\neq 0)$$
respectively.

The other is to define a new probability space. If the original probability space is $(\{0,1,...\},\Sigma, P)$ then define a new space $(\{1,2,...\},\Sigma', P')$ such that, for $A\subseteq \mathbb{N}-\{0\}$, $P'(A)=\frac{P(A)}{1-P(\{0\})}$ and $\Sigma'=\{S-\{0\}\ |\ S\in\Sigma\}$. Then we can define $Y$ as the restriction of $X$ to $\{1,2,...\}$.

It's probably easier to go with the first one.

Oh right so $Pr(\textrm{some expression involving }X|X\neq 0) = 1-Pr(X=0)$.

 

[Ray Vickson]

Oh right so $Pr(\textrm{some expression involving }X|X\neq 0) = 1-Pr(X=0)$.

NO, NO, NO!
$$P({\cal A} | X \neq 0)$$
is given by the usual conditional probability formula, which you should know thoroughly by now.

 

[squenshl]

NO, NO, NO!
$$P({\cal A} | X \neq 0)$$
is given by the usual conditional probability formula, which you should know thoroughly by now.

So $P({\cal A} | X \neq 0) = \frac{P({\cal A} \cap X \neq 0)}{P(X \neq 0)}$. Can we use a truncated distribution via $f(x|X>0) = \frac{g(x)}{1-F(0)}$ where $g(x) = f(x)$ for all $x>0$ and $g(x) = 0$ everywhere else and $F(x)$ is the CDF?

 

[Ray Vickson]

So $P({\cal A} | X \neq 0) = \frac{P({\cal A} \cap X \neq 0)}{P(X \neq 0)}$. Can we use a truncated distribution via $f(x|X>0) = \frac{g(x)}{1-F(0)}$ where $g(x) = f(x)$ for all $x>0$ and $g(x) = 0$ everywhere else and $F(x)$ is the CDF?

You tell me.

 

[squenshl]

How about part 4. With respect to the two non-zero examples above, discuss when this adjustment can be ignored, i.e. it has little impact on the what is actually observed.

Are they asking for an example in which $x=0$ is included?

 

[Ray Vickson]

How about part 4. With respect to the two non-zero examples above, discuss when this adjustment can be ignored, i.e. it has little impact on the what is actually observed.

Are they asking for an example in which $x=0$ is included?

No. They seem to be asking for circumstances wherein $E(X|X \neq 0)$, $\text{Var}\,(X|X \neq 0)$ and maybe $P({\cal A}|X \neq 0)$ are almost the same as $EX$, $\text{Var}\,X$ and $P({\cal A})$. I think they want you to define "almost the same", for practical purposes.