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Non-Zero Discrete Distributions

  1. Mar 30, 2016 #1
    1. The problem statement, all variables and given/known data
    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.

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

    3. 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???
     
  2. jcsd
  3. Mar 31, 2016 #2

    andrewkirk

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    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.
     
  4. Mar 31, 2016 #3
    Oh right so ##Pr(\textrm{some expression involving }X|X\neq 0) = 1-Pr(X=0)##.
     
  5. Mar 31, 2016 #4

    Ray Vickson

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    NO, NO, NO!
    [tex] P({\cal A} | X \neq 0) [/tex]
    is given by the usual conditional probability formula, which you should know thoroughly by now.
     
  6. Mar 31, 2016 #5
    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???
     
  7. Mar 31, 2016 #6

    Ray Vickson

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    You tell me.
     
  8. Mar 31, 2016 #7
    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???
     
  9. Mar 31, 2016 #8

    Ray Vickson

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    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.
     
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