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# Probability question watch

1. A random variable X is binomially distributed with parameters n=40 and p=0.15.
Evaluate the probabilities P(X (less than or equal to) 2) and P(X (greater than or equal to) 3) using i) the binomial distribution and ii) the poisson distribution.

I think iv done part i) ok for the 1st one i got 0.04860 using the tables and assume i just do 1-P(X less than or equal to 2) for the second, however I'm confused about how to do it using the poisson distribution as the tables don't go up to n=40.... can anyone give me a hand? THANKS
2. From the binomial distribution,

E(X) = np = 40 * 0.15 = 6, Var(X) = npq ≈ np ( as p is small, and q is close to 1)

as E(X) ≈ Var(X), E(X) = λ

∴ you can aprroximate to the binomial, X ~ Bin(40,0.15) with X ~ Po(6)

EDIT the answers using the poission are, P(X « 2) = 0.061968 and P(X » 3) = 0.938031.
3. How did you get those answers for the poisson distribution as n is too big for the tables?!
4. If X has a binomial distribution, with parameters (n,p) then, if n is large and p is small it can be approximated by the Poisson Distribution: X~ P(np).

In this case, n=40 p =0.15, 40*0.15=6, so X~ P(6). You can then use the Poisson tables with λ=6.
5. Oh sorry I just made a really silly mistake i was thinking i should be looking up n=40 on the tables instead of x... thanks!

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Updated: November 25, 2005
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