Random variables. The law of distribution of a discrete random variable
It is frequently the case when an experiment is performed that we are mainly interested in some function of the outcome as opposed to the actual outcome itself. For instance, in tossing dice we are often interested in the sum of the two dice and are not really concerned about the separate values of each die. That is, we may be interested in knowing that the sum is 7 and not be concerned over whether the actual outcome was (1, 6) or (2, 5) or (3, 4) or (4, 3) or (5, 2) or (6, 1). Also, in coin flipping, we may be interested in the total number of heads that occur and not care at all about the actual head-tail sequence that result. These quantities of interest, or more formally, these real-valued functions defined on the outcome space, are known as random variables.
A random variable is understood as a variable which as result of a trial takes one of the possible set of its values (which namely – it is not beforehand known).
We denote random variables by capital letters of Latin alphabet X, Y, Z, …, and their values – by the corresponding small letters x, y, z, ….
Example. The number of the born boys among hundred newborns is a random variable which has the following possible values: 0, 1, 2, …, 100.
Example. The distance which will be flied by a shell at shot by a gun is a random variable. Really, the distance depends not only on installation of a sight, but also from many other reasons (force and direction of wind, temperature, etc.) which cannot be completely taken into account. Possible values of this variable belong to some interval (a, b).
Example. Suppose that our experiment consists of tossing 3 coins. If we let Y denote the number of heads appearing, then Y is a random variable taking on one of the values 0, 1, 2, 3 with respective probabilities
A discrete random variable is a random variable which takes on separate, isolated possible values with certain probabilities. The number of possible values of a discrete random variable can be finite or infinite.
For a discrete random variable X, we define the probability mass function p(a) of X by
p(a) = P(X = a)
A continuous random variable is a random variable which can take all values from some finite or infinite interval. Obviously, the number of possible values of a continuous random variable is infinite.
The most full, exhaustive description of a random variable is its law of distribution.
Any ratio establishing connection between possible values of a random variable and probabilities corresponding to them refers to as the law of distribution of the random variable.
About a random variable speak that it «is distributed» under the given law of distribution or «subordinated» to this law of distribution.
For a discrete random variable the law of distribution can be set as a table, analytically (as a formula) and graphically.
The elementary form of assignment of the law of distribution of a discrete random variable X is a table (matrix) in which all possible values of a random variable and the probabilities corresponding to them are listed in ascending order, i.e.
| x1 | x2 | … | xi | … | xn |
| p1 | p2 | … | pi | pn |
or
Such a table is called the series of distribution of a discrete random variable.
The events X = x1, X = x2, …, X = xn, consisting in that as a result of trial the random variable X will take on values x1, x2, …, xn respectively, are incompatible and uniquely possible (because in the table all possible values of a random variable are listed), i.e. form a complete group. Hence, the sum of probabilities is equal to 1. Thus, for every discrete random variable
(This unit is somehow distributed between values of a random variable, therefore from here the term "distribution").
A series of distribution can be represented graphically if values of a random variable are postponed on the axis of abscissas, and on the axis of ordinates – their corresponding probabilities. Connecting the received points forms a broken line named a polygon of distribution of probabilities.
Example. 100 tickets of a monetary lottery are released. One prize in 50 roubles and ten prizes on 1 rouble are played. Find the law of distribution of a random variable X – cost of a possible prize for an owner of one lottery ticket.
Solution: Write the possible values of Х: х1 = 50, х2 = 1, х3 = 0. The probabilities of these possible values are those: р1 = 0,01; р2 = 0,1; р3 = 1 – (р1 + р2) = 0,89.
Let's write the required law of distribution:
| X | |||
| P | 0,01 | 0,1 | 0,89 |