Lecture 1. Discrete random variables
The theme of today's lesson "Random variables".
Most of the time will be given to a particular case of random variable - discrete random variables, but first we’ll introduce the concept of a random variable
A random variable is a function, which as a result of the experience takes a certain value, that is, when conducting experiments (e.g., toss the dice), as a result we can get a certain number (e.g., the fallen out face of the dice) - it will be a random variable .
There can also be introduced other random values.
Symbol Ωx (omega index xi) denotesa set of all possible values of random variable x.
Here we must always be careful not to confuse a set of the random variable and a set of all elementary outcomes of some experience, because it is not always the same.
Random variables can be divided into three types:
1) Discrete random variables are random variables, when a set of values is finite or countable (countable means that they can be enumerated by natural numbers).
2) Random variables are called continuous if a set of value of this random variable is an interval or a set of intervals.
3) Mixed. All others will be mixed random variables, we will not discuss them, that is, in our course, we will focus only on discrete and continuous random variables.
Let’s consider a few examples.
The experience is throwing the dice, then we have a set of values Ωx = {1,2,3,4,5,6}, these are the values of a random variable x.
If as a random variable we take not the number of the fallen out points, but the number of fallen out sixes, then the random variable will take values0 or 1. (0 if it is not six fallen out, 1 if six is fallen out)
Different experience.
Let the coin be tossed till the headfalls out.
We have already considered that, when we studied repeated independent experiments.
Then, as a random variable x, we can, for example, consider the number of tosses of a coin till the first head.
We may be lucky at once (i.e., we did at least one toss), or not, only after the second toss or so on.
That is, there can be arbitrarily many failures,a set of values of such random variable is a set of natural numbers.
More examples.
Let the point be placd on segment [a, b], at the same time, we believe that any point of the segment can be received with equal probability.
If we consider the random variable –the coordinate of the given point, the set of values of this random variable will be a segment from a to b.
Please note that if in previous cases, a set of values is finite or countable, in this case, a set of values is an interval.
Thus, this random value will not be discrete, and it will be continuous.
Consider a second example for a continuous random variable.
Let a device work, and we want to understand how much time will pass before its first stop (that is, until the moment when it is broken in any way, fails somehow and so on).
We assume that at the moment of start the device cannot break down, and it will take some little timebetween the switch-on time and the failure time (it can be very little, but it will exist), the device can break down after any time, or it can work indefinitely.
Thus, the set of the random variable is the segment (0; + ∞), in some cases, we can assume (-∞; + ∞) values of the random variable.
When we consider the continuous random variables, it will take the entire number axis.
When we talk about discrete random variables, then we will set themby a distribution series.
Let's first understand what it is.
Since a discrete random variable has a finite or countable sets of values, that is, its valuesare discrete, we assign a probability to each value.
We have introduced a discrete probability space at the second lecture.
If each value is compared toa probability we will get a table, which is called the "distribution series" (see. Video).
The distribution series arranges the values in ascending order, the second line shows the desired probability.
Let us understand that.
The random variable took the value of x1, random variable took the value of x2, and so on, took a random variable value xn.
These events will form a complete group of events, as it cannot accept several values at the same time, while it is to take at least one of these values, which means that the sum of these events will be a persistent event and the probability of such event is equal to one.
But as these events are mutually exclusive, then the probability of the sum of these events is equal to the sum of their probabilities, and therefore, p1 + p2 +…+pn is equal to one (see. Video).
Consider an example.
The experience is again tossing the dice, and we don’t consider a standard random variable, but the number of sixes fallen out.
Then the random variable can take the values0 or 1, as we have said.
The probability that it will take the value of 0 means the probability that six will not fall out- this probability is 5/6.
Accordingly, the probability that six falls out will be equal to 1/6.
We verify that 5/6 + 1/6 is equal to one.
The distribution series is shown in the slide (see. Video), so we just recorded these values in the table.
If we look at some event, and the random variable is the number of times the event occurred in one experiment, such a value is called a random event indicator.
But there are still other names for this value, sometimes it is called Bernoulli distribution, sometimes it is referred to as alternative distribution, so if you meet in the literature other names, do not panic.
Consider another example.
The dice is thrown more than once, but as many as n, and we will also watch a six fallen out.
That is, random variable x is the number of sixes fallen out, then it can take the value from zero to n (Ωx = {0,1,2, ..., n}), or all the times six fell out or never, or some times it fell out, some times did not.
Then remember that the probability of the random variable can be calculated by Bernoulli's equation, since tossingthe dice is nothing else but simply repeated independent experiments.
Thus, we came to the binomial distribution.
If we carry out n independent experiments, where the probability of success is p, and the random variable that we consider is the number of successes in n experiments, then we can say that our random variable has a binomial distribution with parameters n and p.
This is indicated by letter B, and the parameters are indicated in brackets: B(n, p).
Then the set of values Ωx = {0, 1, 2, ..., n}, and the probabilities are calculated by Bernoulli equation (see. Video).
Limit to the binomial distribution is a Poisson distribution.
Random variable x has a Poisson distribution with parameter λ (lambda) (i.e. one parameter is denoted p(λ), pis a Greek letter), if its set of values is the set of non-negative integers, and probabilities are calculated by formula P(x = k) = e-λ ∙ λk / k! (See. Video).
It is easy to show that if in Bernoulli equation n tends to infinity, and np to λ, then probabilities of Bernoulli equation will tend to probabilities, calculated according to this formula.
It is quite easy to prove, I suggest you to do it yourselves.
And the last distribution, which we will look at is a geometric distribution.
We also have encountered with it when talking about random events.
Suppose, again, we carry out repeated independent experiments, but now we as a random variable we’ll take the number of experiments until the first success.
This random variable has a geometric distribution, denoted by G (p), where p is the probability of success.
The set of values is the set of natural numbers, probabilities of this random variable will be calculated as P(x = k) = p∙qk-1, where p is the probability of success, q is the probability of failures, x= k is random variable x taken value k.