[SOLVED] CS Poisson Random Variables

Poisson Random Variables
EX BIN5 Compare P(X = 5) for the Binomial n = 1000, p = 0.0035 to the appropriate Poisson.
EX 26 The # of butterflies X spotted on a walk across a field is a Poisson random variable with parameter . Given a butterfly, the probability it is a monarch is p, with the type (monarch or not) of different butterflies independent.
What is the distribution of the # of monarchs spotted?
If = 12 and p = 0.2, find the probability of spotting a) exactly 1 butterfly b) more than 5.
Notation: o(h).
A function is f(x) is o(x) (small o of x) if lim f (x) = 0.
x0 x
Examples: f(x) = x2 is o(x). So is g(x) = x3. f(x) = x is not o(x). Nor is g(x) = sin x, nor h(x) = 1.
Suppose f(x) is o(x) so that lim f (x) = 0 . Suppose also that g(x) is o(x): lim g (x) = 0 .
x0 x x0 x lim f(x)+g(x)=lim f(x)+limg(x)=0+0=0
Consider f(x) + g(x). Since
h0 x h0 x h0 x
we have that f(x) + g(x) is then o(x). Similar for a f(x) and therefore f(x) g(x) (take a = 1) as well as for the product f(x) g(x). All these would be o(x). However, the quotient would not necessary be o(x). If f(x) = x2 and g(x) = x3 then g(x) / f(x) = x which is not o(x); nor in this case
is f(x) / g(x) = 1/x. lim1 x = lim 1 which does not exist. x0 x h0x2
The idea is that an o(x) function becomes negligible faster than does x (faster than linear) as x tends to 0.
EX SERVE The number of requests to a server is a Poisson process with rate 50 per minute. Whats the probability that in a 5-second interval there are
a) two requests?
b) more than ten requests?