Question

3. Consider a discrete random variable X taking values k = 0, 1, 2, ... with probabilities P(X = k) = ?^k/k! e^-?, by using ?_{l=0}^k (k over l) = 2^k. where ? > 0. This is the Poisson distribution with parameter ?. This exercise illustrates that the sum of independent Poisson variables again has a Poisson distribution. a. Let X and Y be independent random variables, each having a Poisson distribution with ? = 1. Show that for k = 0, 1, 2, ... P(X + Y = k) = 2^k/k! e^-2, by using ?_{l=0}^k (k over l) = 2^k. b. Let X and Y be independent random variables, each having a Poisson distribution with parameters ? and ?. Show that for k = 0, 1, 2, ... P(X + Y = k) = (? + ?)^k/k! e^{-(?+?)}, by using ?_{l=0}^k (k over l) p^l (1 - p)^{k-l} = 1 for p = ?/(? + ?). We conclude that X + Y has a Poisson distribution with parameter ? + ?.

          3. Consider a discrete random variable X taking values k = 0, 1, 2, ... with probabilities
P(X = k) = ?^k/k! e^-?,
by using
?_{l=0}^k (k over l) = 2^k.
where ? > 0. This is the Poisson distribution with parameter ?. This exercise illustrates that the sum of independent Poisson variables again has a Poisson distribution.
a. Let X and Y be independent random variables, each having a Poisson distribution with ? = 1. Show that for k = 0, 1, 2, ...
P(X + Y = k) = 2^k/k! e^-2,
by using
?_{l=0}^k (k over l) = 2^k.
b. Let X and Y be independent random variables, each having a Poisson distribution with parameters ? and ?. Show that for k = 0, 1, 2, ...
P(X + Y = k) = (? + ?)^k/k! e^{-(?+?)},
by using
?_{l=0}^k (k over l) p^l (1 - p)^{k-l} = 1
for p = ?/(? + ?).
We conclude that X + Y has a Poisson distribution with parameter ? + ?.

3. Consider a discrete random variable X taking values k = 0, 1, 2, ... with probabilities
P(X = k) = ?^k/k! e^-?,
by using
?l=0^k (k over l) = 2^k.
where ? > 0. This is the Poisson distribution with parameter ?. This exercise illustrates that the sum of independent Poisson variables again has a Poisson distribution.
a. Let X and Y be independent random variables, each having a Poisson distribution with ? = 1. Show that for k = 0, 1, 2, ...
P(X + Y = k) = 2^k/k! e^-2,
by using
?l=0^k (k over l) = 2^k.
b. Let X and Y be independent random variables, each having a Poisson distribution with parameters ? and ?. Show that for k = 0, 1, 2, ...
P(X + Y = k) = (? + ?)^k/k! e^-(?+?),
by using
?l=0^k (k over l) p^l (1 - p)^k-l = 1
for p = ?/(? + ?).
We conclude that X + Y has a Poisson distribution with parameter ? + ?.

Added by Madison G.

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