Why Does This Rule Hold: $P(X>F)=1-P(X<F-1)$?

Question

I'm studying for my first statistics exam and I've been trying to calculate $P(X>5)$ for an event which is Poisson distributed. The Poisson distribution is defined as $P(X<5)$ in table in which I want to look up the correct numbers. I thought that $P(X>5)$ would be equal to $1-P(X<5)$ but the right answer says $P(X>5)=1-P(X<4)$. How come this is true? Does this hold for all distributions or is it only defined for Poisson/binomial distributions?

Answer 1

I think your statistical tables are not strict inequalities. I.e., they are giving the probabilities $P(X\le x)$ rather than $P(X<x)$.

Making this adjustment, the thing to note is that the Poisson distribution is discrete, i.e. it only takes whole number values, like the number of cars going over a bridge in a certain amount of time. So $P(X \ge 5)$ means the probability that $X$ can take values $5, 6, 7, \ldots$, and all the way upto infinity (because the Poisson is unbounded).

What are the values the Poisson can take that aren't $5, 6, 7, \ldots$? These are the value $0, 1, 2, 3, 4$. So $P(X\ge 5) = 1 - P(X\le 4)$.

It is also possible to have a continuous probability distribution, which isn't restricted to whole numbers, like the height of a child in a class of schoolchildren. In this case you do have $P(X>x) = 1 - P(X<x)$ and in your specific case $P(X>5) = 1 - P(X<5)$. (Also, $P(X\ge 5) = 1-P(X\le 5)$, because with continuous probability distributions the chance of having any exact value is zero.)

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Edit: your question also mentions the binomial distribution. Like the Poisson, the binomial distribution is discrete. However, it's bounded, so $X$ can be any whole number between $0$ and some maximum $n$. A typical example for the binomial is the number of heads in $n$ coin tosses.