We continue the study of common distributions with the so-called Poisson Distribution, for which we give an intuitive derivation and point out a relation with the binomial distribution. We then turn our attention to continuous random variables. These can take on an uncountable number of values and are characterized by their probability density functions $f_X$ which allow one to compute $$ P \bigl( X\in [a,b]\bigr)=\int _a^b f_X(x)\, dx, $$ for $-\infty\leq a\leq b\leq\infty$. They can be used to determine the random variable's expected value and variance, for instance. We will consider the uniform, normal, and exponential distributions.
The Poisson distribution is used to estimate the number of events recorded in a window of time. It is defined through its probability mass function $$ p(k)=\frac{\lambda ^k}{k!}e^{-\lambda},\: k\in \mathbb{N}, $$ which depends on a parameter $\lambda >0$, which represents the average number of occurrences per interval (fixed as a reference). An intuitive derivation for the distributions aimed at giving an insight into its nature is offered.
Examples of Poisson distributed quantities are considered. Linearity of expectation $$ E(X+Y)=E(X)+E(Y)\text{, where $X,Y$ are discrete random variables,} $$ is established. Generic functional properties of a cumulative distribution functions $F_X$ are presented:
Here we do actually introduce continuous distributions, or, by extension, continuous random variables via their probability density function as described above. Their expectation and variance can be computed by $$ E(X)=\int_{-\infty}^\infty x\, f_X(x)\, dx\text{ and }\operatorname{var}(X)=\int _{-\infty}^\infty (x-E(X))^2\, f_X(x)\, dx $$ respectively. Think about the relation/connection between the formulæ for expectation and variance of discrete and continuous random variables. Can you see the parallels?