Expected Length of a Binomial Proportion?

Question

I was reading this article (https://www.scirp.org/pdf/ojs_2021101415023495.pdf) here and came across the following equations (page 7):

Coverage Probability of a Binomial Random Variable:

$$Cp(p,n) = \sum_{k=0}^{n} I(k,n) \binom{n}{k} p^k (1-p)^{n-k}$$

Expected Length of the Interval:

$$EL(p,n) = \sum_{k=0}^{n} \binom{n}{k} p^k (1-p)^{n-k} [ U(x) - L(x)]$$

I am interested in learning how I can derive these two formulas.

For instance, suppose if I start with some Random Variable "x" that has a Binomial Probability Distribution Function (https://en.wikipedia.org/wiki/Binomial_distribution). I understand that :

The "Coverage Probability" (https://en.wikipedia.org/wiki/Coverage_probability) refers to probability of the Confidence Interval containing the parameter of interest
And the "Expected Length" (https://stats.stackexchange.com/questions/18379/expectation-of-length-of-a-confidence-interval, Estimate length of confidence interval) refers to the length of the Confidence Interval relative to the Variance

But how can I use this probability distribution function to derive these two formulas starting from the Binomial Probability Distribution Function?

Thanks!

Apply the Law of the unconscious statistician: $\mathbb{E}f(X) = \sum_{k=0}^n f(k)\binom{n}{k}p^k(1-p)^{n-k}$, when $X\sim\text{Binom}(n,p)$. — user51547, Jan 02 '23 at 03:19
Your $EL$ is missing ${n \choose k}$ from the original article while I am not particularly happy with $U(x)-L(x)$ as I would have thought they should be $U(k,n)-L(k,n)$ — Henry, Jan 02 '23 at 03:27
@ Henry: thank you for pointing this out! I now made this correction! — stats_noob, Jan 02 '23 at 03:32

Expected Length of a Binomial Proportion?

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