m200-Ch04-1

Hello and Welcome to Math 200 Statistics Online here Kenyatta College.

My name is Ray Lappas.

In this video we will take a quick dive into section 4.1

where we study the probability distribution

functions for discrete random variables.

First of all, let's recap what a discrete random variable is.

First of all, a random variable describes the outcome of a statistical experiment.

Now these outcomes could be qualitative or quantitative.

And the values of the variable can vary within each repetition.

So we would run an experiment collect the information and so the information that

we're collecting would be the random the general name for a random variable.

As a notation. We use upper case letters such as X. And Y.

To denote a random variable in general.

And then lower case would describe the specific value of a random variable.

Uh

They give an example here where they let X be the number

of heads you get when you toss a coin three times.

And so if you toss a coin three times. They list down the possibilities here the T. T.

T. Stands for tales for all the three tosses that you had the second th H.

Is the scenario where we toss the tails first

and then the heads and then the heads again.

And that should be different from the third scenario

where you toss ahead Tales and a head etcetera.

So if you look at these possibilities,

how many possible uh tosses that you get would where X would be the number of heads.

That would come out. So in the first scenario the T. T. T. There are no heads there.

So that would be zero. So one possible value for our random variable would be zero.

And then further along the line

the H. T. T. The th T. And the T th all have one heads in it. Uh and two tails.

And so there are three of those scenarios but uh we're counting the exes there so X.

Is equal to one in that particular case.

Uh And then you just go on and count the rest of the information.

So that is a reminder of what of random variable is.

And since we are talking about discrete random

variables are outcomes for our X values.

Should just be whole numbers.

In other words we're not gonna get any fractions and it's not qualitative data

where we're not gonna be expected to look at words for these things.

Now a discreet probability distribution is a a

scenario or distribution that has the characteristics where

each of the probability that you get uh is going to be between zero and one.

And we know that probabilities are between zero and one.

So that should be pretty straightforward.

But the idea here is that the probability distribution

you have a collection of all your X.

Values

and then you have an assigned probability to your X values.

And those assigned probabilities if you add them all up should equal to one.

So the sum of all the probabilities is equal to one.

So the idea here is that you get you get an X. Value for your random variable.

And then for each of these X values for your random variable,

you're gonna get assigned a probability.

And those probabilities are of course gonna be between zero and one.

And then when you add up all those probabilities you should get one.

So here's an example.

Child psychologist is interested in a number

of times that uh newborn babies crying,

wakes the mother after midnight

and here's a random variable uh where X is going to be equal to the number

of times a week and newborn babies crying wakes up its mother uh after midnight.

So it could be zero uh crying baby might not wake up the mother at all, or crying,

baby might wake up a mother once and in 2345 et cetera.

So X is going to be the number of times that uh that

the that the mother wakes up after they hear the baby crying.

And then for each of those values there is an assigned probability. So the capital X.

Um would represent the the general

random variable.

And the little excess that they're showing here would represent the actual value.

So X is equal to zero. There's a probability of two out of 50 X is equal to one.

There's a probability probability of 11 out of 50 X

is equal to there's a probability of 23/50 et cetera.

Okay, so this is a random variable

and this is a discrete random variable because the X.

Values are all um all whole numbers.

So the probabilities are all between zero and one and if we add

them all up you should get the whole thing to equal to one.

Okay so each of the p of excess are between zero and one

inclusive and the sum of all the probabilities is equal to one.

Alright let's look at another example,

suppose nancy has uh classes three days a week attends.

She attends the class three days a week um 80% of the time.

And she only goes twice on the two days 15% of times.

And there's a chance that she only makes it one of the days at 4% of the time.

And there's a possibility that she doesn't go to class at all.

And that would be uh 1%,,

suppose the one week uh

suppose one week is randomly selected. Uh So X.

Is equal to the number of days uh that nancy attends class

so she can either attend class 3 to 1 or zero D.

Days that particular week

X takes on the value of what So remember X.

Is gonna be the random variable that takes on the values of whatever X.

Is to be describing. And that's the number of days that nancy attends class so that is

321 or 00123.

And I suppose one week is randomly chosen construct.

The probability distribution called the pdf table uh shown

in like the previous example that we saw.

So here are the results.

So x is going to be the number of days that nancy attends a class per week.

And then the possible values for X. R 012 and three.

And then here's your probably distribution table

based on the scenario that they gave.

So she will attend or she might attend zero days and that happens 1% of the time.

She could attend one day the week in the whole week.

And that's, that would happen 4% of the time.

She would attend two days per week at 15% of the time. And

She would be attending all four days or all three days of that of her classes,

80% of the time.

Okay,

So that's the quick scenario, quick overview of Section 4.1,

where we start studying PDFs for discrete random variables.

Okay, I hope that helps

m200-Ch04-1

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