Introduction to Discrete Probability Distributions - Probability and Statistics, Mathematics Video Lecture - Engineering Mathematics

we've already spent some time looking at
random variables and defining random
variables in different ways and the one
that I keep alluding to may be because
it's one of the simplest definitions of
a random variable is one that's derived
from a coin flip maps the outcomes of a
coin flip to some numbers to a random
variable and so we could say that the
random variable is equal to 1 or say so
let's say it's equal to 0 if we get
tails on my coin and let's say if it's
equal to 1 if I get heads on my coin now
the one thing that this random variable
definition is not doing for us so far is
telling us the probability of getting
the random variable to be 0 or 1 and to
get that information we need something
called a probability distribution
probability probability distribution for
this random variable essentially tells
us the probability that we get any one
of these outcomes now is this a discrete
or a continuous random variable well the
outcomes here are distinct they are
countable you could list them this is
definitely a discrete random variable so
we will construct or we will be taking a
look at a discrete probability
distribution this Griet probability
distribution so let's say someone says
ok I've defined this random variable in
some way for a given coin and I'm going
to give you the probability distribution
and so they come here and they start
drawing this probability distribution
and most probability distributions they
tend to be depicted visually and so here
in the horizontal axis I will plot all
of the possible outcomes so you can view
this as kind of the outcomes for my
random variable X so the outcomes are
either I'm going to get a 0 or 1 so 0 or
1 there's no other outcomes I guess you
could list other outcomes and just say
they have a 0 probability of happening
but these are the ones that can actually
happen and then here on the vertical
axis I'll prop plot the probability of
getting any one of those outcomes and so
the person tells you ok the probability
of getting a 0 is 50% 50% so they draw a
little bar here
it's like a histogram a bar chart and so
this goes right up here to 50% which
I'll draw as a decimal 0.5 and they tell
you the probability of getting a one of
the random variable becoming a one is
60% 60% draw it just like that 60 60 %
or 0.6 now my question to you and I know
you're just starting to be exposed to
probability distributions but based on
what you know about probability is this
a legitimate probability distribution
well let's think about what the
probability of all of getting any any
one of these outcomes so what let's
think about let's think about the
probability the probability that our
random variable is equal to zero or
we're doing a union here not an and or
the probability that our random variable
is equal to one these are all of the
possible outcomes we're essentially
saying what's the probability that we
get one of the possible outcomes for our
random variable well over here we would
just add the two probabilities so the
probability of this one right over here
is 0.5 and then we're taking a union so
and these are mutually exclusive events
so we can just add the two together and
then we have the probability of getting
a 1 the probability giving a 1 is 0.6
actually let me do that in a different
color than blue the blue is 0.6 and this
strangely adds up to 1.1 somehow saying
that you have a hundred and ten percent
probability of getting any of the
possible outcomes this makes absolutely
this makes absolutely no sense you
cannot have a greater than 100 percent
chance of having any one of the outcomes
if you're looking at all of the outcomes
from a random variable it should add up
to 100% so you point this out to the
gentleman who's trying to tell you that
this coin is defined in this way or the
probabilities are and he says all yes
yes yes yes I made a mistake that was
not the probability distribution this
right over here is not 0.5 0.5 let me
clear it
actually let me clear this in redraw it
so he realizes that this was actually
0.4 0.4 and now my question again to you
is this does this at least look like a
legitimate probability distribution well
sure
the the sums of the exclusive outcomes
that this random variable can take on
now equal to one point six point four
and now the last thing I'll ask you
given what you know about this random
variable is clearly an unfair coin are
you more likely to get heads or tails
well the random variable you're clearly
more likely to get an outcome of one if
you look at the definition of a random
variable the random variable will be one
when your coin is heads so it's clearly
somehow the coin is weighted a little
bit more probable to having heads so
I'll leave you there in the next few
videos we'll study more and more
probability distributions which is
really the core of probability and you
see it becomes a very helpful thing when
we start to study inferential statistics