## Continuity Correction

When a random variable is discrete, its frequency function is viewed as the height of relative frequency histogram with interval . Then we can approximate its probability by the area under the curve.
Example.
Suppose that has a binomial distribution with `n = 15` and `p = 0.3`.
We can find the probability
on the interval "`intval = c(2,7)`."
The exact probability (in black) and the normal approximation (in red) with continuity correction
are shown in the display.

n = 15 p = 0.3 intval = c(2,7) nn = 0:n pmf = dbinom(nn,n,p) mean = n * p sd = sqrt(n * p * (1-p)) x = seq(-0.5, n+0.5, length=100); y = dnorm(x, mean, sd); plot(x, y, type='l', lwd=2, frame.plot=F, main="Normal Approximation"); prob.hist(nn,pmf,lty=1) nn = intval[1]:intval[2] pmf = dbinom(nn,n,p) prob.hist(nn,pmf,lty=1,col='yellow') x = seq(intval[1]-0.5, intval[2]+0.5, length=50); y = dnorm(x, mean, sd); polygon(c(x,max(x),min(x)), c(y,0,0), lwd=2, col=2, density=20); prob = pbinom(intval[2],n,p) - pbinom(intval[1]-1,n,p) text(intval[1], pmf[1], pos=4, round(prob,digits=4)) prob = pnorm(intval[2]+0.5, mean, sd) - pnorm(intval[1]-0.5, mean, sd) text(intval[1], pmf[1], pos=2, col=2, round(prob,digits=4))

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