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I just read that the cumulative distribution function for a binomial random variable is a "step function where the function is flat and then jumps at each nonnegative integer value".

Can someone explain to me why the CDF is a step function with horizontal lines between each integer intervals and not simply points with probabilities at each integer?

Thanks.

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    $\begingroup$ The cumulative distribution function $F(x)=\mathbb P( X \le x)$ is typically a function $\mathbb R \to [0,1]$ whether the underlying distribution is continuous, discrete, singular or some mixture. If it was instead $\mathbb Z \to [0,1]$ then it might be a collection of points, but it would then be less useful. $\endgroup$
    – Henry
    Commented Oct 7, 2020 at 9:55

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The CDF is defined the probability that your random variable takes a value less than or equal to a real number:

$$ F\colon\mathbb{R}\to\mathbb{R}, x\mapsto F(x):=P(X\leq x). $$

And of course if $X$ only has probability mass on the natural numbers, the probability that $X$ is (e.g.) less than or equal to $3$ is the same as the probability that it is less than or equal to $3.3$ or $3.8$. Which is precisely the same as saying that the CDF is flat between $3$ (inclusive) and $4$ (exclusive) and only jumps at the natural numbers.

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