So, I'm currently studying introduction to statistics and I'm on the section about Discrete Random Variables and their distribution tables.
Some questions ask to find the probability within certain ranges (e.g: within one standard deviation from the mean, less than one standard deviation from the mean, etc.)
Since they are discrete variables, we can't have decimals in our $P(x)$ function. But I'm confused on how to round before adding up the probabilities.
For examples, $P(7.22<x<15.99)$, would that be between $7$ and $16$? or $7$ and $15$? And would it make a difference if it was $P(7.22\le x\le 15.99)$? And what about $P(7.99<x<15.22)$?
Thank you in advance!
Best Answer
In a practical situation, in order to formulate the limits of probabilities such as $P(7.22 < X < 15.99)$ for a random variable $X$ that takes only integer values, you would need a good reason for picking the exact endpoints 7.22 and 15.99. (Why not endpoints 7.23 and 16.00?)
However, once formulated, there can be no question that $$P(7.22 < X < 15.99) = P(7.22 \le X \le 15.99) = \sum_{i=8}^{15} P(X = i).$$ That is, the sum of eight individual probabilities. For example, the binomial random variable $X \sim Binom(20, 1/2)$ has positive probability only at integer values. It makes no difference what non-integer values are included between the stated endpoints. In particular, for this binomial random variable, we have $P(7.22 < X < 15.99) = 0.862503,$ computed in R as follows.
Of course, one could also use the binomial PDF formula to compute each of the eight terms, and then add them together.
I hope it is now clear that $P(7.22 < X < 16.00)$ and $P(7.22 \le X \le 16.00)$ have different values, unless it happens that $P(X = 16) = 0.$