Is This a Poisson or Binomial Random Variable?

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The discussion centers on distinguishing between Poisson and Binomial random variables in a mathematical context. The user derives the Poisson probability mass function as ((λ)^x)/((e^(2λ))*x!) for y=1 and attempts to manipulate the equations to identify a Binomial distribution. The challenge lies in demonstrating that the variable x, which ranges from 1 to infinity, conforms to a Binomial distribution, particularly when using the relationship P(X=x and Y=y) = P(X|Y)P(Y).

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  • Understanding of Poisson and Binomial distributions
  • Familiarity with probability mass functions
  • Knowledge of conditional probability
  • Basic calculus, particularly exponential functions
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  • Study the derivation of the Poisson distribution in detail
  • Learn how to apply the Binomial theorem in probability
  • Explore conditional probability and its applications in statistics
  • Investigate the differences between discrete and continuous random variables
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Students in statistics, mathematicians, and anyone involved in probability theory who seeks clarity on the distinctions between Poisson and Binomial random variables.

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1. http://d.imagehost.org/t/0866/problem1.jpg



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3. For the Poisson random variables I get ((lamda)^x)/((e^2lambda)*x!) when y =1. Is this a Poisson Random variable? Also, when y = 0, I get ((lamda)^x)/((e^lambda)*x!) - ((lamda)^x)/((e^2lambda)*x!) I cannot get the binomial. I don't know how to show that this is Binomial as x is in the range from 1...infinity.

For part 1 I used the fact that P(X=x and Y=y) = P(X|Y)P(Y)
and P(Y=y) = sum(1 to x)P(X=x and Y=y) = 1, so P(X=x|Y=y)(P(y) = P(X=x and Y=y)
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