Intro to Discrete Random Variables - Online Tutor, Practice Problems & Exam Prep

Intro to Random Variables & Probability Distributions

Welcome back, everyone. So in previous videos, we've talked a lot about the different probabilities of certain events happening. We've also talked a lot about frequency distributions. We're basically gonna put those ideas together in this video, and we're gonna start talking about these things called random variables and probability distributions. This is really, really important, and we're gonna be talking a lot about these in the next couple of videos.

So I want to walk you through some key concepts and information you need to know, and then we'll do an example together. Let's get started. What is a random variable? A variable is just a letter that stands for a number. It's exactly what's going on here.

And the random part just means that it represents a single number determined by chance for each outcome of what we call an experiment. For example, let's say you go out and you enter a random raffle, and these are the number of prizes that you could win in that raffle. These are numbers, and they're determined entirely by chance. There's no skill involved. Alright?

Now there are basically two different types of random variables that you'll need to know. One is called a discrete random variable (DRV), and these are basically when the numbers cannot be broken down further. The classic example I like to use is a dice roll. Your outcomes are either 1, 2, 3, 4, 5, or 6. I can't roll a dice and get something like a 2.4.

That doesn't make any sense. Right? So that is a discrete random variable. The other kind is called a continuous random variable (CRV). This is basically the opposite.

These numbers can be broken down further. If you go out and you survey a bunch of people and measure their heights, you don't just get 72 or 75 inches. You can get all the numbers in between. And if you had powerful enough instruments, you could get into the decimals and things like that. You can break those numbers down even further. Alright?

So why is this all important? Well, basically, what's gonna happen in these problems is you're gonna be doing a bunch of experiments, and you'll organize the outcomes of that experiment with their associated probabilities. And this is what's called a probability distribution. It's basically a table, just like a frequency distribution, and it shows the probabilities of all the possible values that a random variable can be. Alright?

So here's how I like to think about this. It's actually very similar to a frequency distribution. We would go out and survey a bunch of random people. Let's say it was the number of cans of soda they drank per day. Here were your classes, and you would tally up all of their frequencies.

This is something that you're doing sort of after the fact. A probability distribution shows all of the different outcomes of an experiment and their probabilities. And usually, you use these things to predict the theoretical outcomes of this before it happens. That's kind of the idea here. Alright?

Now one of the very first things that you might have to do with a probability distribution is just verify that the sum of all probabilities is 1, and that's exactly what we're gonna do in this first example. So let's go ahead and get started here. So we're gonna verify this table meets the criteria of a probability distribution, and there are basically two criteria that you need to know. One of the things you'll notice is that all of these probabilities are decimals instead of percents. That's usually how you're gonna see them.

And, basically, the first criterion is that for any outcome of this experiment, so for any value of x, remember that's just your random variable, the probability has to be a number between 0 and 1. You can't have an outcome that represents a 110% probability because that doesn't make any sense. Alright? So that's the first criterion. And if you go ahead and look through these numbers, all of these numbers are just decimals that are between 0 and 1.

So that's the first criterion, and this is definitely met. What about the second one? The second one is that when you add up all of the different possible probabilities for all of the outcomes, that sum, remember that's big sigma over here, has to be 1. So, in other words, it has to be a 100%. Think about it.

If I enter this raffle and all of my outcomes are either 0, 1, 2, 3, or 4, that represents all the possible outcomes and therefore the probability has to be a 100% for any of those things or all of those things combined happening. Alright? So, basically, all of these numbers have to be between 0 and 1, but they have to add up to 1. Alright? So let's go ahead and make sure that this second criterion is met.

Just go ahead and add up all the probabilities, 0.10, 0.20, 0.40, 0.20, and then 0.10. Alright? If you go ahead and add all of these things up, what you should find is that this is a number that equals 1. So in other words, this is a probability distribution. If you added up all of these numbers and you did not get 1, you got something that was significantly less or greater than, then this would not be a probability distribution.

Alright? So that's basically it for probability distributions. Let's go ahead and take a look at our second example here. Alright? So we're gonna be asked to play a random lottery, and we're going to pay $1 to enter this lottery, and we have the profits and probabilities organized. So in other words, these are the outcomes, and these are the associated probabilities. Okay? So let's take a look at the first question here.

What is the missing probability in the table? You'll notice that in this column, one of these numbers is actually missing over here, and we actually don't know what that is. Can we actually find that? Well, the idea here is that, remember, all of the different outcomes have to add up to a 100%. So all of the outcomes of the probabilities, right?

That's the sum of p_x. Okay? So what we can use here is if all of these things have to add up to 1, oh sorry. All these have to add up to 1, right?

100%, which equals 1. So in other words, basically, what happens is the probability that you have a $5 profit is basically just going to be 1 minus everything else. Right? It's going to be 1 minus all of the other probabilities. 0.40 plus so this is actually going to I'm going to put this in parentheses.

A student is analyzing different types of variables in a statistics class. Which of the following below is a discrete random variable?

• A. The time it takes for a randomly selected runner to complete a 5k race
• B. The weight of a randomly chosen bag of apples from a grocery store
• C. The number of defective lightbulbs from a randomly chosen batch in a factory
• D. The number of days in a random month

Intro to Random Variables & Probability Distributions Example 1

Welcome back, everyone. Let's take a look at this example here. So we have in a random survey, people are asked how many sodas they drink per day. I want to talk about this because we've actually referenced this in a previous video.

In a previous video, we said that if you were to go out and do a random survey of the number of sodas drank per day, you could create something like a frequency distribution. You just tally up all of the frequencies for these certain classes. However, you could also use this as a probability distribution. Basically, what you could use this for is to predict the outcome that if you were to go grab another random person out of that survey, they would respond with one of these seven outcomes, and the probabilities are listed here. So, you could use this information as a frequency distribution, or in this case, we're going to use this as a probability distribution.

Let's take a look at the first question here. What's the probability that you pick another person out of this random survey, and they respond with having at most 2 sodas per day? So what does "at most" mean? It means, basically, starting at 0 all the way up until and including 2. So whenever you see "at most," it just means that it is a probability.

So I'm going to write this: "At most" means that it's the probability that x ≤ 2. Alright? So in other words, what you're going to do is start at px=0, and you're going to add up everything all up until and including x=2. Right?

That's "at most." So, basically, you're just going to grab these first three rows, 0, 1, 2, and you're just going to add those probabilities together. So I'm just going to add these 3 up. Alright? So I've got 0.50 plus 0.31 plus 0.09.

If you go ahead and work this out, what you're going to get here is you're going to get 0.90. So in other words, if you grab any random person out of the survey, the probability that they would have at most 2 sodas per day is 90% or 0.90. Usually, we're going to express these things as decimals. Alright?

Let's take a look at the second question. What's the probability that a person, again, you pull a random person now out of the survey again, that they're going to respond with having between 1 and 4 sodas per day. What does "between" mean? Well, now what's going to happen is that, basically, instead of "at most" 2, we're just going to be looking at these two numbers between 1 and 4. So "between" 1 and 4, again, just means that we're going to add up a bunch of probabilities. First, you're going to start with px=1. You're going to add up everything up until and including x=4. Okay?

So I'm going to be looking at these 4 entries now between 1 and 4. Okay? So I've got 0.31 plus 0.09 plus 0.05 plus 0.03.

What you should get out of this is you should get 0.48. So in other words, there is a 48% chance that if you grabbed a random person, they would respond between 1 and 4 sodas. Alright? So that's how you do these kinds of problems.

Let me know if you have any questions. Hopefully, you got that, and thanks for watching.

Expected Value (Mean) of Random Variables

So, one of the very first things we learned how to do when given a dataset was to calculate the mean. And, basically, we just added up a bunch of values and divided by the total number of observations to get a number, like 1. Well, now we've been talking a lot about probability distributions, which are a little bit different. Here, we have outcomes of an experiment with their associated probabilities. But nonetheless, one of the things you may have to do is calculate the mean of a random variable.

So I want to show you how to do that in this video. And basically, we're going to use an equation where we're going to be adding a bunch of things, but there's one other operation we have to do first. I want to show you how to do that, and we'll do an example together. Let's get started here. So, to find the mean of a discrete random variable, how do we do that?

Why can't we just add up a bunch of these numbers and divide by the total number? For example, let's say I had a probability distribution where I would count up the number of kids in a random household for a random town. And let's say this was my probability distribution: 0, 1, or 2, with their associated probabilities. So why can't I just add up these numbers, 0, 1, and 2, and divide by 3? Well, it's basically because each of these numbers can't be treated as equals.

Each one has their associated probabilities, which are different. For example, one kid in a household would be the most common at 60%, whereas zero kids would be 15%. So you can't treat these things as equals. Alright? So then how do we find the mean?

Well, basically, what we do is we're going to have to multiply each value, in other words, each x, by its associated probability p(x). So you have to do x times p(x), and then that's when you can add up everything. So as an equation, to find μ, you basically you're going to still do a σ, which is to add up everything, but you're going to have to do x times p(x) first. Now, by the way, one of the things that you may see in a course or a book is the expected value. Basically, that means the exact same thing.

Whenever you're asked to find the expected value of a distribution, that just means find the mean. So let's go ahead and do that for this specific example here. One of the things I want you to notice here is that a lot of times the data that you'll be given in these types of problems is going to list the outcomes and probabilities horizontally like this. What I want you to do is if you're not given a table like this one, I want you to make one where the outcomes are listed vertically, and then you basically have columns like this for the probability. And then I want you to make one for x times p(x).

This makes it much easier to read off your calculations. For example, if I want to figure out x times p(x), so I can add up everything, I could just go horizontally across 0 times 0.15. Alright? So, 0×0.15 = 0 because 0 times anything is just 0. 1×0.60 equals 0.60. And then finally, 2×0.25 becomes 0.50.

Okay? So now I have all my x times p(x)'s. Remember, this equation says once you get all the x times p(x)'s, you have to add up everything. Okay? So basically, just gonna add up everything straight down this column, and that's gonna be my expected value E[X].

Alright? So if I just go ahead and add up everything, what I'm gonna notice here is that this is a number 1.1. This would be my μ or E(X). Those things mean the exact same thing. Alright?

So that's my answer. And there are two things I want to talk about here. Right? One of them well, first of all, this is a number, my expected value, that isn't one of my outcomes. It's not 1, or 2, and that's perfectly fine.

So you might be asking, how do I get 1.1 kids? Basically, what this number represents, the way that you interpret it, is if I were to sample, let's say, a 1000 or 10000 households, count up all the number of kids, and then take an average or a mean, my average would be around 1.1. That would be about roughly my mean. Alright? So even though it's not one of my outcomes and I can't get 1.1 kids, if I were to do a large enough sample and then take the mean, that would be roughly how many average kids I have per household.

Okay? So that's what that number means. And the second thing I want to discuss is that this number tends to be close to the number that has the highest probability. Alright? Because, again, if you were to take a bunch of random samples of this, the most likely outcome that you're gonna get is a bunch of ones. And so when you average out all that data and take the mean, it's gonna be a number that's close to this number over here. Alright? So that's it for how to calculate our mean of a discrete random variable. Now that we understand this, let's go ahead and get some practice. Thanks for watching.

A factory produces lightbulbs in batches of 50. The probability distribution for the number of defective lightbulbs in a randomly selected batch is shown below. Find the expected value.

• A. 0.17
• B. 1.7
• C. 0.03
• D. 2.5

Variance & Standard Deviation of Discrete Random Variables

Welcome back, everyone. So in a previous video, we saw how to find the variance and standard deviation of a dataset, which just basically represents how spread out the data values are. But now we've been talking about discrete random variables and probability distributions, but you may have to find the variance and standard deviation for those. Conceptually, it means the same exact thing. Basically, how spread out are the numbers, the outcomes of your experiments.

But because we're dealing with probabilities, the equation is a little bit different. I'm going to show you how to calculate these in this video, and, basically, it's just going to be this equation over here. It looks a little scary at first, but I'm going to show you that all that happens is that when we're given a discrete random variable, we're going to make a table with columns for \( x \times p(x) \), then \( x^2 \times p(x) \). Alright? Let's go ahead and just jump right into this example, and I'll show you how this works.

Alright? So again, these equations look kinda scary at first, but basically, all that this equation is saying is you're going to take each value, each \( x \), square it, multiply it by its associated probability, add all those things up, and then we're going to subtract the mean squared. Alright? This bottom equation is something that you may see in a book. It's a little bit more conceptually easy to understand.

You take the difference between each value and the mean, then square that, add all those things up, and multiply by each probability. Alright? So it's a little bit different, but, ultimately, you'll get the same exact answer. Personally, I find this first equation is easier to use. That's the one we're going to use in this problem.

Alright? So let's just jump right into our problem. These are numbers we've already seen before. This table here shows the probability distribution for the number of kids that you would find in a random household in a random town. We're going to find the variance and standard deviation of this distribution here.

Okay? So again, when we have these data values that are given to us horizontally like this, the first thing you want to do is you want to build a table with the values listed vertically. It's much easier to do it this way. We've already seen actually all of these numbers before, so we're not going to recalculate them. But, basically, if you were to just say each \( x \times p(x) \), you would get this column over here, and you would already even find the mean, which was the expected value, which is 1.1.

Alright? We actually already know what the mean of this distribution is. The mean is just 1.1 or 1.10, which means that we can also find the mean squared, because we just square that number. So, basically, this is just going to be 1.21 if you calculate that. Alright?

So if you look at this equation here, this giant mess over here, we actually already know what this number is. We already know what the \( \mu^2 \). So, really, all that happens here is we just have to figure out what is the \( x^2 \times p(x) \). And, again, that's why you want to build a little column here for \( x^2 \times p(x) \). Okay?

So I'm just going to take each number and square it and then multiply it by its probability. So we've got \( 0^2 \times 0.15 \), which is still just 0. Then I've got \( 1^2 \times 0.60 \), which is still 0.60. Then I've got \( 2^2 \times 0.25 \), which is going to equal, and that's going to equal 1.

Great. So if you add up all of these things here inside of this column, what you should get is you should get 1.60. So, basically, what happens is this represents the sum of \( x^2 \times p(x) \). Right? That's basically what this number represents.

That's actually what goes right here in this column. So you should make these little rows here. It's going to help you organize this information. So this is going to be 1.60. Okay?

So now how do you find the variance? Well, basically, what happens is you're going to take this thing over here and you're just going to subtract this thing over here. So we're going to take 1.60 like this, and we're going to subtract the previous number of 1.21. That's going to give us a variance of 0.39. Right?

This is basically this number minus this number. That's how you find the variance. Alright? So that's our first answer. We got 0.39.

And the last question we have to answer is what's the standard deviation? And remember, whenever you have one of these things, you should always figure out the other because they're just related by squares and square roots. The variance is just the standard deviation squared. So we just basically take the square roots of this number we just found over here. What you should get is you should get a standard deviation of 0.62.

You may see this rounded to the 1st decimal place just because all of our data values are integers. So you may see this rounded to 0.6. And, basically, what happens is this has units, the standard deviation over here. The standard deviation would be 0.6, you know, kids per household or something like that. Okay?

So these are your two answers. You've got a variance of 0.39 or 0.4, and then we've got 0.6 kids. Alright? So, basically, what happens is, again, this represents the spread of the outcomes of this experiment. We've got a mean of 1.1, that's the expected value, with a standard deviation of about 0.6 kids.

Alright. That's it for this one. Now that we understand a little bit more about this, let's go ahead and take a look at some practice. Thanks for watching.

A company tracks the number of complaints they receive, where the random variable X is the number of complaints received daily. Find the variance & standard deviation of this distribution.

• A. Variance = 0.83; Standard Deviation = 0.9
• B. Variance = 0.9; Standard Deviation = 0.83
• C. Variance = 0.83; Standard Deviation = 0.85
• D. Variance = 0.85; Standard Deviation = 0.9