Binomial Random Variable With Non-Integer value

[-Job-]

So the problem gives a binomial random variable X with parameters n=5 and p=0.25 and ask for the probability P(X=1.5). The binomial probability mass function is defined only for integers. Should i approximate using the normal distribution or the poisson?

 

[Dale]

I would say that the probability would be 0. I don't think an approximation is really warrented here since n is not very large and p is not very small. Besides, to get a probability with a continuous PDF you would have to pick a range of X, not just X=1.5.

-Dale

 

[-Job-]

Yeah, that's what i was about to say, since n isn't large there isn't any guarantee that the binomial distribution will behave normally. I hate trick questions. I'm guessing that, if the questin asked P(X<=1.5) for the same parameters, then it would be equivalent to asking P(X<=1) right?

 

[Dale]

Yeah, it makes sense to ask about the CDF for non-discrete values, it just looks like a stair step type function as you described. It makes sense, but it still isn't very nice :)

-Dale

 

[-Job-]

Ok so, the next question gives a Poisson random variable X with lambda = 1/3 and asks for both P(X<=2.5) and P(X=-2). I'm guessing the same trick applies to the first one and is equivalent to asking P(X<2) since a poisson distribution function is discrete and defined only at integer values.
The second one though is what, 0 again? I can't see that a poisson random variable can have negative values because its probability mass function contains a factorial which isn't defined for a negative value. What's up with these questions?

 

[Dale]

Seems like they really want to emphasize the "discreteness" and the domain of some of these functions. But it feels like they are trying to emphasize the fact through trick questions. In any case, you are correct, the Poisson is not only a discrete distribution it is also defined only for non-negative numbers.

-Dale