Now we are moving into the next concept that is called a Random variable.
That means you have a probability space, started with the probability space and you are defining
a real valued function which maps omega to R such that if you find out the inverse image
of any x, the real line, that inverse image between
minus infinity to x belonging to F. If this condition is satisfied by any real valued
function which maps from omega to R, then that is going to be called it as a random
variable.
That means after you have a collection of possible outcomes you are finding one sigma
algebra. You can make more than one sigma algebra over the omega. So you have a one
fixed sigma algebra, it could be trivial one or the non trivial one and so on. So you have
a fixed F, after fixing the F you have a probability measure and the probability measure is nothing
to do with the random variable at all.
Still you have a probability space and from the probability space you are defining a real
valued function, such that inverse image is belonging to F, that I can make out with the
simple diagram.
This is the omega and from the omega you have created the F. F means it has the events and
the events are nothing but the few possible outcomes. That means these possible outcomes
you will land up with the one element and this possible outcome you will land up with
another event and so on. So like that the different few possible outcomes that is going
to be one of the elements in the F.
So you have created another real valued function that is X, from omega to R, this is a real
valued function and you take any point some x in the real line and you find out what is
the inverse image from minus infinity till x you collect what is the inverse image you
got it under the mapping X from minus infinity to the closed interval x. You collect all
the possible outcomes that is going to give the value between minus infinity to closed
interval x, you collect such a possible outcome.
If you collect such a possible outcomes and that is going to be one of the elements in
the F, for different values of x then the real valued function is going to be called
it as a random variable. That means once you know the F, if you create a real valued function,
after checking that condition you can conclude that real valued function is going to be a
random variable.
That means, if you have some other F there is a possibility some real valued function
may not be the random variable. That means how you choose F, that is going to play a
role of come to the conclusion the real valued function is going to be a random variable
or not. If you take F is going to be the largest one, the largest sigma algebra over F, over
omega then any real valued function is going to satisfy this property.
Suppose you take the sigma algebra over omega which is in between the trivial one and the
largest one then the few real valued function may be a random variable and few other real
valued function may not be the random variable. So in the usual scenario whenever you see
the random variable definition in many books, they use real valued function is going to
be a random variable just like that, that means they have taken, the F is going to be
the largest sigma algebra.
So whenever F is going to be the largest sigma algebra then any real valued function is going
to be a random variable. And going back to the previous slide, this condition is going
to be the if and only condition also. Suppose we have a, real valued function is going to
be a random variable then this condition will be satisfied and if this condition is satisfied,
then that real valued function is going to be random variable also.
Now we are moving into the next concept called cumulative distribution function. So the cumulative
distribution function for the random variable X can be defined as, capital F suffice x is
for the random variable X and the small x is the variable x that is going to be probability
of X is less than or equal to small x. And here the x lies between minus infinity to
infinity. So this is going to be called it as CDF of the random variable X.
So the way I relate with the probability of X is less than or equal to x, this X is less
than or equal to x is nothing but you collect few possible outcomes such that, under the
operation X of w that gives the value less than or equal to x for all w belonging to
omega. That means you collect a few possible outcomes w such that under the mapping X of
w should give the value maximum x. That is less than or equal to, therefore this is nothing
but an event. And this event is belong to the capital F.
Therefore, the way you have taken the probability of X is less than or equal to x, therefore
as the x moves from minus infinity to infinity, you are keep on including some more possible
outcomes over the x. Therefore, the probability of X is less than or equal to x it varies
over the x, you are going to get more probability values, so therefore this F of x is going
to satisfy few properties.
So if you see the properties of F of x, this values is always lies between zero to one
for all x. Suppose you take x is almost minus infinity then it is going to be zero and it
is towards the infinity then it is a going to be one. That means I make out limit x tends
to minus infinity, this is going to be zero and the limit x tends to infinity, the F of
x is going to be one.
The third property, the way we are keep on accumulating the possible outcomes and trying
to find out the probability and that too you make it as F of x. Therefore, the F of x is
monotonically increasing function in x. That means over
the x if you take two values x is less than or equal to y, then the F of x value will
be less than or equal to F of y, that means as x is less than y either it takes the same
value or greater than value.
Therefore, it is going to be in the way it is called monotonically increasing function
in x. The fourth one, it is going to be right continuous function in x. That means either
it is going to be a continuous function, if it is not a continuous function it is a right
continuous, that means the left limit exists for any x as well as the right limit exists
and either it is going to be a left limit is same as the right limit and value defined
at that point.
Or the left limit is different from the value defined at that point which is equal to the
right value. Therefore, the function is going to be call it as a right continuous. So the
CDF is going to be a continuous function or it is going to be a right continuous function.
I can show few diagrams of CDF, as the x goes, the F of x will start from zero and will land
up, one. So this is going to be a F X continuous as well as it satisfies the condition of minus
infinity to infinity, it is going to be zero, minus infinity it is zero, infinity it is
one and it is monotonically increasing and continuous function and I can give another
example of CDF. It starts from zero and it has a discontinuity.
So that means it has the, it is a right continuous function and monotonically increasing function
and it has the countably infinite jumps or countably infinite discontinuity and it reaches
at infinity one. So based on the way the CDF goes I can give one more example in which
this is going to be continuous in some, then it has the jumps. That is also possible.
So the way the CDF is going to be a continuous function from minus infinity to infinity or
the CDF is going to have the countably finite jumps or countably infinite jumps or it has
a both type then you can classify the random variable as a discrete random variable, continuous
random variable or mixed type random variable. So the random variable is going to be called
it as discrete type random variable if the CDF is going to have a countably finite or
countably infinite jumps in the CDF, then it is called the discrete random variable.
If any random variable has CDF, has continuous function from minus infinity to infinity then
that random variable is called it a continuous random variable. If any random variables CDF
has both continuous between sum interval and countably finite or countably infinite jumps
in some interval, then that random variable is going to be called it as a mixed type random
variable.