Frequency Function

When an outcome is a numerical value, the outcome determined by an experiment is referred as random variable (r.v.). We conventionally denote random variables by uppercase letters $ X, Y, Z, U, V, \ldots$ from the end of the alphabet. Particularly if the random variable $ X$ takes ``discrete'' values such as 1,2,3,$ \ldots$ (or 0,1,2,3,$ \ldots$), we call it a discrete random variable. The statement such as ``$ X = 2$'' is an event, and therefore, is associated to the probability $ P(X = 2)$. We can assign $ P(X=j)$ for all the possible values $ j = 1,2,3,\ldots$ (or $ j = 0,1,2,3,\ldots$), which will completely describe the ``probabilistic nature of the random variable $ X$'', that is, the probability distribution of $ X$. $ P(X=j)$ is often called a frequency function, and simply written by $ P(j)$.

The frequency function $ P(i)$ is graphically presented. The bar graph may be able to indicate a particular outcome with the highest frequency, the existence of mode of the distribution of interest.

To find the probability of an event from the frequency function $ P(i)$, it is important to determine the exact range $ \{j \le X \le k\}$ of the random variable $ X$. Then it is computed as

$ P\big($ $ \le X \le$ $ \big) = P(j) + P(j+1) + \cdots + P(k) =$

Essential parameters of discrete distribution are the mean $ \mu$ and the standard deviation $ \sigma$. They are computed as follows.

$ \mu = \displaystyle\sum x \cdot P(x) =$ ; $ \quad\sigma = \displaystyle\sqrt{\sum (x - \mu)^2 \cdot P(x)} =$