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?

  • $\begingroup$ Hint: Use the Complementary event rule. $\endgroup$
    – user29102
    Aug 13, 2013 at 13:43
  • 3
    $\begingroup$ Are you sure you have this right? For an inequality $P(X\ge 5) = 1 - P(X \le 4)$. For a strict inequality $P(X>5)$ means the probability that $X$ takes values $6,7,8,9\ldots$, which is the same as $1$ minus the probability of X taking $0,1,2,3,4,5$. So, $P(X>5) = 1-P(X<6)$. $\endgroup$
    – TooTone
    Aug 13, 2013 at 13:44

1 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.)

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.