I am teaching the normal distribution as a continuous random variable and my students have just learned binomial distribution. Wondering what would be a good idea to explain the transition from discrete to continuous where, instead of histogram, we now find the probability as area under the curve?
Would appreciate any suggestion. Thanks.
On
I think the best way for an introductory probability course to make the transition from discrete to continuous begins by explaining the Poisson process, first showing that the number $X_t$ of arrivals before time $t$ is so distributed that $$ \Pr(X_t=x) = \frac{(\lambda t)^x e^{-\lambda t}}{x!}. $$ Then let $T_x$ be the time until the $x$th arrival, and show that the two events $$ \big[ X_t < x \big] \quad \text{and} \quad \big[ T_x >t \big] $$ are the same event and therefore have the same probability. From this we get $$ \Pr(T_1 > t) = \Pr(X_t =0) = e^{-\lambda t} \text{ for } t\ge0 $$ and there you have a continuous probability distribution.
Fix the mean of the binomial, some $M$, and then let $p = M/n$ and consider the sequence of distributions $$\mathcal{B}(n,p) = \mathcal{B}(n,M/n).$$
You will converge to a continuous distribution, you can show this by the sequence of pdfs.
Another way is to fix any distribution $\mathcal{D}$, even discrete, and find $$S_n = \frac{1}{n} \sum_{k=1}^n X_k$$ where $X_k \sim \mathcal{D}$ for all $k$. Then $S_n$ will converge to the normal distribution (by the CLT).