Continuity correction: Change P(2 ≤ x < 9) to continuous?

Question

Convert discrete probability into continuous probability using continuous correction:

attempt:

Discrete: P(2 ≤ x < 9)

therefore continuous should be

Continuous: P(1.5 < X < 8.5)

Is this right? or should it be P (1.5 < x < 9)?

Answer 1

I will assume that your discrete random variable takes integer values.

I find it difficult to remember a bunch of rules, so I remember only one: That if $k$ is an integer, and we are approximating the discrete $X$ by a continuous $Y$, then $\Pr(X\le k)$ is often better approximated by $\Pr(Y\le k+0.5)$.

Now $\Pr(2\le X\lt 9)=\Pr(2\le X\le 8)=\Pr(X\le 8)-\Pr(X\le 1)$.

We are ready to apply the only rule that I remember. Our probability is approximately $\Pr(Y\le 8.5)-\Pr(Y\le 1.5)$. (Of course, for the continuous $Y$, there is no difference in probability between $\le$ and $\lt$.)