A key statistical concept is that of random variable, which is understood as the numerical result of a random experiment and it is so called because precisely the result is unknown a priori, or in other words, it is the result of chance.
Good examples of these kinds of experiments are coin and dice tosses (performed honestly), because the result of a particular toss is not known until it is done..
For example, simultaneously tossing two coins once, or tossing a coin twice, could have the following results, denoting the appearance of a head as C and a seal as S:
Many variables can be defined for a random experiment, for this one in particular the "number of heads" could be defined, and its result is entirely random..
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The usual way to denote random variables is through the last two letters of the alphabet: X and Y, in uppercase. In this way, continuing with the example of coins, the random variable X can be defined like this:
X = number of heads obtained in a simultaneous toss of two coins.
This variable can take the following numerical values: 0, 1 and 2 and each of them has an associated probability of occurrence. The set of these probabilities is known as probability distribution and indicates the possible values of X and the way to assign the probability to each.
Probability distributions can be given in the form of a graph, table or even a formula.
Some are very important and are studied assiduously, because many random variables adhere to them. For n honest coin tosses, the distribution of the experiment is called binomial distribution.
Random variables can be of two types:
It is important to distinguish between one type and the other, since the form of treatment of the variable depends on this..
Discrete random variables are characterized by being countable and assuming very specific, determined values. In the toss of the two coins, the random variable X = number of heads obtained in a single throw, is discrete, since the values that it can take are 0, 1 and 2 and no other.
Also the result of rolling two dice is a random experiment, in which discrete random variables can be defined, such as this:
Y = "the sum of both tosses is 7"
A 7 can be obtained as a sum by six different possibilities of the first die and the second die:
The set of favorable results for the event of obtaining a 7 can be summarized as follows:
(1,6); (2.5); (3,4); (4.3); (5, 2); (6.1)
The probability that any of these events occurs is 1/6, since according to the classical definition of probability, there are 36 possible outcomes, of which 6 are favorable to the event in question:
P (get 7) = 6/36 = 1/6
More examples of discrete random variables are:
Although in these examples the values of the variables are natural numbers, something very common, it should be noted that discrete random variables can also take decimal values.
Continuous random variables take infinite values, without jumps or gaps between them, so unlike discrete random variables, which are countable, continuous ones are said to be uncountable.
So to represent continuous variables, an interval is used, for example the interval [a, b], within which all the possible values of said variable are found.
An example of a continuous random variable is the amount of milk a cow gives per day. Between the value considered minimum and the maximum, for example in milliliters, a cow can give any amount of milk per day.
For these variables, the probability distribution is a function called a function probability density.
In the following examples of random variables, there are discrete variables and there are also continuous ones. To know what variable rate it is, it is necessary to specify if the variable in question is countable or not, since this is the characteristic that differentiates the discrete variables from the continuous ones..
This is a discrete random variable, whose values are the natural numbers with 0 included. It is known to be discrete, not because its values are integers, but because they can be counted, even if the count results in very large numbers..
Indeed, it may be that on the appointed day to count people, not a single one uses the subway, although it is not the most likely. In this case, the random variable is worth 0, but surely many people will travel in the subway.
Assuming that N people traveled that day, the random variable "X = number of people who use the subway in one day" takes integer values between 0 and N.
This is also a discrete random variable. The maximum value that it reaches is the total number of students enrolled and the minimum is 0, if on the day the count was carried out, no student was able to attend class.
For example, assuming that the class has a total of 25 students enrolled, this random variable assumes the values:
0, 1, 2, 3… 25
On a farm there are a certain number of cows, some are small and weigh less, others are large and weigh more. Between the cow with the lowest weight and the cow with the highest weight, there is a whole range of possibilities for the weights of a cow chosen at random, therefore it is a discrete random variable.
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