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