Discrete Random Variables (Part 1)

追加: 2026-05-17

[00:00:00]hi guys so we're moving on to discrete probability distributions we learned probability in the previous chapter and now we're going to start building our distribution using probabilities okay we're going to start by learning what a random variable is and essentially we're going to be able to take a qualitative variable and turn that into a quantitative variable okay let's start by an experiment all right so we're gonna use flipping a coin as our little experiment okay so what kind of things can we get from flipping a coin twice twice right so let's define our sample space here right so we can get two heads we can get two tails we can get heads and tails or we can get tails and heads right so we have four outcomes in our sample space now how I have this written these are qualitative variables now quantitative variables and with qualitative variables we really cannot do much as far as calculation goes because these are letters right and then we really cannot do addition subtraction multiplication division with words so what we could do is this let's say we're interested in counting how many tails we get by flipping a coin twice so we could define a capital X to be a number of tails right so we could count the number of tails rather than listing it out as now outcome so if we go through our outcomes here this one has zero tails and this one has two tails right and this one has one tails this one has one tails right so our options are sample space for this capital X will be x equals 0 1 & 2 no tails 1 tails and two tails ok so when we do this now we've turned this qualitative variable right into a quantitative variable and this X right here okay this X this one right here okay is actually called a random variable so if the outcome of a probability experiment is a numerical result then we say that outcome is a random variable all right so let's read the definition random variable is a numerical measure important numerical measure of the outcome from a probability experiment so it's value is determined by chance random variables are denoted using letters such as capital X so in our coin flipping example the random variable X is the number of tails in two flips of a coin and the possible values of the random variable X are 0 1 & 2 now our particular example of flipping a coin and counting the tails was a discrete random variable because it had a countable or finite number of values another way of looking at this is that we can plot the values of a discrete random variable on a number line in there will be space between each point so x equals 0 that means 0 tails right 1 tails 2 tails 3 tails 4 tails like that a continuous random variable on the other hand has infinitely many values so when we plot those on a line right it's just a continuous line like this so there is no spaces in between the values right since a value of a continuous random variable can take on any number between an interval all right so let's look at some examples example 1 determine whether the following random variables are discrete or continuous state possible values for the random variable question 8 the number of is earned in mrs. Forbes statistics class with 50 students enrolled okay so the random variable here is the number of A's earned in my class so X is the number of ADEs right okay because we can count those right it is a discrete random variable I'm going to put RV for random variable so what are the possible outcomes for this one well it is possible that no one gets an A in my class oh that would be really sad okay but yeah that could happen and it is possible that one person gets an A or it is possible the two people get an A or three students get an A and so on right and could it go all the way up to 50 by 50 because we have a total of 50 students enrolled right so all 50 students could get an A right okay so X will be 4 will go from 0 to 50 all right those are the possible outcomes of the random variable ok question B the speed of the next car that passes a state trooper okay so the random variable for this one be the speed okay and it is a continuous continuous random variable why because speed is measured right now the possible values are well you can say X is greater than zero you can say the possible values are all positive real numbers right now since we don't know the exact upper bound for this we can just leave this as an open-ended interval I mean we know that something like 500 miles per hour does not really make sense with this but just leave it as open-ended an interval is fine okay questions see and I would like you guys to try this one on your own so go ahead and pause the video and try it and when you're done come back and check your answer okay go ahead and check your answer so this is a discrete random variable and the possible values are 0 1 & 2 3 it could go all the way up to whatever number so you can just make it an open-ended and just like it before okay but this one is discrete random variable so you have to list numbers like 0 1 2 like that alright okay so the important thing to remember is that the actual value of x is determined by chance right so for example we don't know the probability of two cars that come to the drive-through it's up to people it's determined by chance so all of these different values look at these different values right here okay have a different probabilities so we organize the information using table graph or formula now probability distribution provides the possible values of the random variable X and their corresponding probabilities a probability distribution can be in the form of a table graph or mathematical formula okay so it follows two rules so you're going to have check two things first you want to make sure that each probability must be between zero and one so each probability must be between 0 1 okay and the second rule if you add up all the probabilities you should get one okay so parts add up to the whole right all right now notice that we have a probability distribution which is a distribution right so whenever we have a distribution it will look at three things what are they well the shape that's the first one shape of distribution and we look at the central tendency such as mean and median mode no so much okay we focus on mean and median all right and the spread spread okay of the data which is a standard deviation or the IQR okay let's look at an example of a probability distribution in the form of a table all right the table below shows the probability distribution for the random variable X where X represents a number of movies streamed on Netflix each month so it looks like the value of a random variable range is from 0 movie to 5 movies a month well obviously this is a little outdated data and I think people sometimes stream 5 movies a week or more these days right but this is what it is okay all right so it's a probability so let's ask ourselves these questions is each probability between 0 & 1 well we can just do a quick scan here right first one point zero six this will be the probability that someone will stream zero movies a month alright so 6 percent probability that the person strings 0 movies right ok and next one is 0.5 t8 58 percent twenty two point twenty two sorry point ten 0.03 0.01 good they're all between zero and one perfect no negative numbers no numbers more than one perfect all right second rule do all the probabilities add up to one well we can check over the first one first of all okay over this one was okay right okay the second one go ahead and add these numbers up guys do the other up to one point zero six plus point 58 plus point twenty two and you're going to add all the way up to point zero one this is after one go ahead and check yep all right so what we can do with this is something really similar to what we did in the frequency distribution chapter so once we had a frequency distribution we made a frequency histogram right so now that we have our probability distribution we can make a probability histogram okay and it is created the same way the horizontal axis corresponds to the value of the random variable so in our case number of movies streamed okay and the vertical axis corresponds to not frequency but the probability that each value of the random variable will occur okay so let's draw a probability histogram for this alright so the horizontal axis again number of movies streamed it goes from zero to five chaos DS right okay and the probability when you look at the table here and the highest probability is point 58 okay so how many do we have here one two three four five six okay so how about we make each tick mark point one so 0.1 0.2 0.3 0.4 0.5 0.6 there we go that would work right okay we already so four discrete variables the number is normally right in the middle of the bar and the probability that they stream zero movies right it's point zero six so it's a little over halfway on the first notch so it's gonna be somewhere here okay right so this is for x equals zero okay it's point zero six point zero six okay all right and how about when x equals one that's point 58 okay so that's almost all the way to the tippy top okay so this is point 58 okay all right so how about two the probability is point twenty two so it's gonna be somewhere here like that okay and three is point ten okay and four point zero three okay all right and point 0.01 for 500k it's really skinny okay all right there we go okay so now we have our distribution so we can talk about the shape right what shape is this yes you guessed it skewed right nice cute right distribution okay so what I wanna do now is I want to focus on this rectangle right here this one okay this rectangle okay alright so I'm gonna bring this one down and we're gonna talk about the area of this rectangle now what is the width of this rectangle here well these discreet Val variables appear they're kind of like a midpoint okay kind of like a midpoint here okay so we're going hero 2 1 at 1 to 2 2 to 3 and so on right so the width is 1 and what is the height here the height is 0.12 so the area of this rectangle is one times one times point when t2 which is point when t2 okay which is the same as the probability of the value of the random variable right here right because the width was one so the area is the same as the probability which is the height of this rectangle so let's write that down is actually very important area is the probability the random variable has that value okay area is the probability self okay okay let's continue on with the mean of the probability distribution okay it's the same concept as a weighted mean so here is the formula music x equals the sum of the value of random variable x times the corresponding probability okay so let's do an example by hand okay so we're gonna go back to the Netflix example compute the mean of the probability distribution okay so first you're gonna be multiplying each value of the random variable by the corresponding probability so zero time is point zero six that would be just a zero and one time is point 58 that will be point 58 two time is point twenty two that's point 44 three times point ten point three four time is 0.03 point well and the last one five time is 0.01 so that'd be 0.05 okay so once you get these products then give it up add them all up okay so when you add them all up you should get one point forty nine and what is the units streamed movies right so this is it this is the mean of the probability distribution and we're going to talk about the interpretation in just a second but I want you to focus on the calculation for now okay now let's use our calculator to find the mean okay so take out your calculator all right so first you're gonna type in your date your data in l1 and l2 okay so I already did that so under l1 you type the values of X okay and then under l2 you type the values of the probabilities right okay and once you did that you're gonna go to stat and you're gonna go to calc okay and we want number one 1-var stats' so go ahead and press ENTER on that okay and then you're gonna type sickand you're gonna press sorry press 2nd l1 and then press comma and you press 2nd and then - so you're listing two lists right here l1 comma L - okay and then you press Enter there you go it's a 1.49 that is the mean of the probability distribution okay okay so there is only six possible values here okay so this was fairly easy to calculate by hand but if it's larger then that would be a kind of a tedious to calculate by hand so you're more than welcome to use your calculator on this okay all right so let's see how we can interpret this mean so I want you guys to recall how we interpreted probability in the previous chapter well we used the law of large numbers right so we do the same thing here because news of X the mean right oh I cannot actually write the symbol here this is news of x equals one point 49 and this is from a probability distribution so whenever we have something related to probability we interpret it using the law of large numbers so suppose that we repeat probability experiment many many times well let's say hundred times with outcomes X 1 X 2 X 2 B all the way up to X 100 right so let's write that down suppose a probability experiment is repeated many times say 100 times with outcomes X 1 X 2 X 3 all the way up to X 100 right so X 1 is basically is the number of movies streamed by customer 1 and X 2 is number of movies streamed by customer 2 and so on so you repeat it hundred times now if we calculate x-bar right from that sample meaning the average right you find the average number of movies streamed by 100 customers what do we do we add up all the data values and divide the total by 100 right so we get right so here if we find X bar which is x1 plus x2 plus all the way up to X 100 and we divide that total by 100 right they'll give us X bar then what we expect is that we expect we expect that the value of X bar will be pretty close right pretty close to the value of muse of X which is the mean of the probability distribution up here