Mean and variance of Bernoulli distribution example | Probability and Statistics | Khan Academy

Summary

The video discusses a simple example of a Bernoulli distribution, where a population is surveyed for their opinion on the president. Respondents can either give a favorable or unfavorable rating, and the mean and variance of this discrete probability distribution are calculated. The example demonstrates how to find the expected value and variance using probability-weighted sums, despite the expected value not being a possible outcome. It concludes by introducing general formulas for the mean, variance, and standard deviation in a Bernoulli distribution, setting the stage for further exploration of the binomial distribution.

Takeaways

• 📊 The speaker conducts a full survey of a population's opinion on the president, offering two possible responses: favorable or unfavorable.
• 🎯 The probability distribution is discrete with two outcomes: 40% have an unfavorable view, and 60% have a favorable view.
• 📈 The expected value (mean) of the distribution is calculated by assigning 0 to unfavorable (u) and 1 to favorable (f) views.
• 🔢 The expected value of the distribution is 0.6, which represents a probability-weighted sum of the two options.
• 🙅 No individual can have an actual value of 0.6; individuals will either have a favorable or unfavorable rating (0 or 1).
• 💡 The variance of the population is the probability-weighted sum of the squared distances from the mean.
• 🔍 Variance is calculated using the differences between each outcome (0 or 1) and the mean (0.6), resulting in a variance of 0.24.
• 📐 The standard deviation is the square root of the variance, which in this case is approximately 0.49.
• 🧠 The distribution is skewed to the right, with most individuals having a favorable view.
• 📚 This specific example introduces the Bernoulli Distribution, a special case of the binomial distribution, which is further explained in future discussions.

Q & A

What is the purpose of surveying every single member of a population in this scenario?

The purpose is to gather data on the favorability rating of the president, with the aim of understanding the distribution of opinions within the population.

What are the two options available for the survey respondents?

The respondents can either have an unfavorable rating or a favorable rating for the president.

What percentage of the population had an unfavorable rating according to the survey?

According to the survey, 40% of the population had an unfavorable rating.

What percentage of the population had a favorable rating?

60% of the population had a favorable rating.

How is the probability distribution represented in this scenario?

The probability distribution is represented as a discrete distribution with two values: unfavorable (0) and favorable (1).

What is the expected favorability rating of a randomly picked member of the population?

The expected favorability rating is the mean of the distribution, which is calculated as 0.4 * 0 + 0.6 * 1 = 0.6.

Why is the mean of 0.6 not a value that an individual can actually take on?

The mean of 0.6 is not a value that an individual can take on because each person must choose either a favorable or unfavorable rating, which are represented as 1 or 0, respectively.

How is the variance of the distribution calculated?

The variance is calculated as the probability-weighted sum of the squared distances from the mean. In this case, it is 0.4 * (0 - 0.6)^2 + 0.6 * (1 - 0.6)^2 = 0.24.

What is the standard deviation of this distribution?

The standard deviation is the square root of the variance, which is approximately 0.49.

What does the distribution's skew to the right indicate?

The skew to the right indicates that the distribution is not symmetric and that there is a higher concentration of favorable ratings.

What is the Bernoulli Distribution mentioned in the script?

The Bernoulli Distribution is a discrete probability distribution that takes value 1 with success probability p and value 0 with failure probability q = 1 - p. It is the simplest case of the binomial distribution.

Outlines

00:00

📊 Understanding Population Favorability Ratings

In this paragraph, the speaker explains a scenario where every member of a population is surveyed about their opinion of the president, with only two response options: favorable or unfavorable. After surveying the entire population, 40% give an unfavorable rating, while 60% give a favorable rating. This forms a discrete probability distribution since only two values are possible. The speaker introduces the concept of expected value (mean) and explains how it’s calculated for this distribution, assigning 0 to unfavorable and 1 to favorable. The mean is found to be 0.6, but no individual has this rating; rather, it's an average across the population. This discrepancy between the actual ratings (0 or 1) and the mean (0.6) is explored.

05:03

🧮 Calculating Variance and Standard Deviation in Discrete Distributions

This paragraph introduces variance, which measures how much the data points in the distribution deviate from the mean. Using the same population distribution (40% unfavorable, 60% favorable), the speaker shows how variance is calculated as the probability-weighted sum of squared differences from the mean. The variance is calculated as 0.24, and the standard deviation (the square root of the variance) is 0.49. The speaker also notes that while it's harder to visualize standard deviation in a discrete distribution, it makes sense that the distribution is skewed to the right since more people gave a favorable rating.

Keywords

💡Probability Distribution

A probability distribution is a mathematical function that describes the likelihood of different outcomes in an experiment. In the video, the distribution is discrete, meaning there are only two possible outcomes: a favorable or unfavorable view of the president. The concept of probability distribution helps to quantify how likely each outcome is in a population survey.

💡Discrete Distribution

A discrete distribution refers to a type of probability distribution where the possible outcomes are distinct and separate, as opposed to continuous. In this script, the survey has only two outcomes: favorable (1) or unfavorable (0). This type of distribution simplifies the analysis of binary events like yes/no or success/failure.

💡Mean (Expected Value)

The mean, or expected value, is the average value that would result from repeated trials of an experiment. In the video, the expected value is calculated to be 0.6, representing the favorability rating, even though no individual can be 60% favorable. It’s a theoretical average based on the probability-weighted sum of the possible outcomes.

💡Variance

Variance measures how much the data points in a distribution differ from the mean. In this video, variance is calculated as the probability-weighted sum of the squared differences between each possible outcome and the mean (0.6). It quantifies the spread or dispersion of the favorability ratings from the mean value.

💡Standard Deviation

Standard deviation is the square root of the variance and indicates how much the outcomes deviate from the mean. In the video, the standard deviation of 0.49 shows how spread out the favorability ratings are around the mean. It’s a useful measure for understanding the distribution’s variability.

💡Bernoulli Distribution

A Bernoulli distribution is the simplest case of a binomial distribution, representing a random experiment with exactly two possible outcomes (like success/failure). In this script, the favorable/unfavorable view is an example of a Bernoulli trial, with probabilities 0.6 and 0.4, respectively. The entire video is centered around calculating characteristics of this distribution.

💡Binomial Distribution

A binomial distribution is a probability distribution that summarizes the likelihood of obtaining a fixed number of successes in a set of independent trials, each with the same probability of success. The Bernoulli distribution is a special case of the binomial distribution with only one trial. This concept is introduced in the video to build towards understanding how probabilities are distributed across multiple trials.

💡Weighted Sum

A weighted sum is a summation where each value contributes proportionally to its assigned weight. In this video, the probability-weighted sum is used to calculate both the mean (expected value) and the variance by multiplying each outcome by its probability before summing the results. This allows for more accurate modeling of random variables.

💡Success and Failure

In the context of the Bernoulli and binomial distributions, success and failure refer to the two possible outcomes of a trial. In the video, a 'success' is defined as a favorable rating (with a probability of 0.6) and a 'failure' is an unfavorable rating (with a probability of 0.4). These terms are crucial in binary outcome models like the one discussed.

💡Squared Distance

The squared distance refers to the square of the difference between a given value and the mean, used to calculate variance. In the script, the squared distance is calculated between 0 and 0.6 (the mean) for unfavorable ratings, and between 1 and 0.6 for favorable ratings. Squaring ensures that all deviations from the mean contribute positively to the variance.

Highlights

Surveying an entire population to measure the favorability rating of the president is typically impractical, but hypothetically possible.

The population has two response options: favorable or unfavorable rating.

In this example, 40% of the population has an unfavorable rating, and 60% has a favorable rating.

The probability distribution in this scenario is discrete, with only two possible values.

The mean (or expected value) is calculated as the probability-weighted sum of the possible values of the distribution.

Defining the unfavorable rating as 0 and the favorable rating as 1 allows for calculating the mean of the distribution.

The mean of the distribution is calculated as 0.6, representing the expected favorability rating.

Even though the mean is 0.6, no individual can have a favorability value of 0.6; individuals can only choose 1 or 0.

The mean represents the expected proportion of favorable responses in the population, not an individual outcome.

Variance is defined as the probability-weighted sum of the squared distances from the mean.

To calculate variance, the distances between the possible values (0 and 1) and the mean (0.6) are squared and weighted by their respective probabilities.

The variance is calculated to be 0.24.

The standard deviation of the distribution is the square root of the variance, which is approximately 0.49.

The distribution is skewed to the right, with the mean closer to the favorable rating (1).

This scenario demonstrates a basic case of the Bernoulli distribution, which is the simplest form of a binomial distribution.

Transcripts

Let's say that I'm able to go out and survey every single

member of a population, which we know is not normally

practical, but I'm able to do it.

And I ask each of them, what do you think of the president?

And I ask them, and there's only two options, they can

either have an unfavorable rating or they could have a

favorable rating.

And let's say after I survey every single member of this

population, 40% have an unfavorable rating and 60%

have a favorable rating.

So if I were to draw the probability distribution, and

it's going to be a discrete one because there's only two

values that any person can take on.

They could either have an unfavorable view or they could

have a favorable view.

And 40% have an unfavorable view, and let me color code

this a little bit.

So this is the 40% right over here, so 0.4 or maybe I'll

just write 40% right over there.

And then 60% have a favorable view.

Let me color code this.

60% have a favorable view.

And notice these two numbers add up to 100% because

everyone had to pick between these two options.

Now if I were to go and ask you to pick a random member of

that population and say what is the expected favorability

rating of that member, what would it be?

Or another way to think about it is what is the mean of this

distribution?

And for a discrete distribution like this, your

mean or you're expected value is just going to be the

probability weighted sum of the different values that your

distribution can take on.

Now the way I've written it right here, you can't take a

probability weighted sum of u and f-- you can't say 40%

times u plus 60% times f, you won't get

any type of a number.

So what we're going to do is define u and f to be

some type of value.

So let's say that u is 0 and f is 1.

And now the notion of taking a probability weighted sum makes

some sense.

So that mean, or you could say the mean, I'll say the mean of

this distribution it's going to be 0.4-- that's this

probability right here times 0 plus 0.6 times 1, which is

going to be equal to-- this is just going to be

0.6 times 1 is 0.6.

So clearly, no individual can take on the value of 0.6.

No one can tell you I 60% am favorable and 40% am

unfavorable.

Everyone has to pick either favorable or unfavorable.

So you will never actually find someone who has a 0.6

favorability value.

It'll either be a 1 or a 0.

So this is an interesting case where the mean or the expected

value is not a value that the distribution can

actually take on.

It's a value some place over here that

obviously cannot happen.

But this is the mean, this is the expected value.

And the reason why that makes sense is if you surveyed 100

people, you'd multiply 100 times this number, you would

expect 60 people to say yes, or if you'd summed them all

up, 60 would say yes, and then 40 would say 0.

You sum them all up, you would get 60% saying yes, and that's

exactly what our population distribution told us.

Now what is the variance?

What is the variance of this population right over here?

So the variance-- let me write it over here, let me pick a

new color-- the variance is just-- you could view it as

the probability weighted sum of the squared distances from

the mean, or the expected value of the squared distances

from the mean.

So what's that going to be?

Well there's two different values that

anything can take on.

You can either have a 0 or you could either have a 1.

The probability that you get a 0 is 0.4-- so there's a 0.4

probability that you get a 0.

And if you get a 0 what's the distance from 0 to the mean?

The distance from 0 to the mean is 0 minus 0.6, or I can

even say 0.6 minus 0-- same thing because we're going to

square it-- 0 minus 0.6 squared-- remember, the

variance is the weighted sum of the squared distances.

So this is the difference between 0 and the mean.

And then plus, there's a 0.6 chance that you get a 1.

And the difference between 1 and 0.6, 1 and our

mean, 0.6, is that.

And then we are also going to square this over here.

Now what is this value going to be?

This is going to be 0.4 times 0.6 squared-- this is 0.4

times point-- because 0 minus 0.6 is negative 0.6.

If you square it you get positive 0.36.

So this value right here-- I'm going to color code it.

This value right here is times 0.36.

And then this value right here-- let me do this in

another-- so then we're going to have plus 0.6 times 1 minus

0.6 squared.

Now 1 minus 0.6 is 0.4.

0.4 squared is 0.16.

So let me do this.

So this value right here is going to be 0.16.

So let me get my calculator out to actually calculate

these values.

So this is going to be 0.4 times 0.36, plus 0.6 times

0.16, which is equal to 0.24.

So our standard deviation of this distribution is 0.24.

Or if you want to think about the variance of this

distribution is 0.24 and the standard deviation of this

distribution, which is just the square root of this, the

standard deviation of this distribution is going to be

the square root of 0.24, and let's calculate what that is.

That is going to be-- let's take the square root of 0.24,

which is equal to 0.48-- well I'll just round it up-- 0.49.

So this is equal to 0.49.

So if you were look at this distribution, the mean of this

distribution is 0.6.

So 0.6 is the mean.

And the standard deviation is 0.5.

So the standard deviation is-- so it's actually out here--

because if you go add one standard deviation you're

almost getting to 1.1, so this is one standard deviation

above, and then one standard deviation below gets you right

about here.

And that kind of makes sense.

It's hard to kind of have a good intuition for a discrete

distribution because you really can't take on those

values, but it makes sense that the distribution is

skewed to the right over here.

Anyway, I did this example with particular numbers

because I wanted to show you why this

distribution is useful.

In the next video I'll do these with just general

numbers where this is going to be p, where this is the

probability of success and this is 1 minus p, which is

the probability of failure.

And then we'll come up with general formulas for the mean

and variance and standard deviation of this

distribution, which is actually called the Bernoulli

Distribution.

It's the simplest case of the binomial distribution.