Name: 概率分佈簡介 (Introduction to Probability Distributions)
Uploaded: 2021-01-14T08:57:44.000Z
Duration: 6 min 20 s
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This lecture is going to serve as an overview of what a probability distribution is and

情態副詞

Simply put, a distribution shows the possible values a variable can take and how frequently

Before we start, let us introduce some important notation we will use for the remainder of

Assume that “upper-case Y” represents the actual outcome of an event and “lowercase

y” represents one of the possible outcomes.

One way to denote the likelihood of reaching a particular outcome “y”, is P of, Y equals

For example, uppercase “Y” could represent the number of red marbles we draw out of a

bag and lowercase “y” would be a specific number, like 3 or 5.

Then, we express the probability of getting exactly 5 red marbles as “P, of Y equals

Since “p of y” expresses the probability for each distinct outcome, we call this the

So, probability distributions, or simply probabilities, measure the likelihood of an outcome depending

on how often it features in the sample space.

Recall that we constructed the probability frequency distribution of an event in the

We recorded the frequency for each unique value and divide it by the total number of

Usually, that is the way we construct these probabilities when we have a finite number

If we had an infinite number of possibilities, then recording the frequency for each one

becomes impossible, because… there are infinitely many of them!

For instance, imagine you are a data scientist and want to analyse the time it takes for

Any single compilation could take anywhere from a few the milliseconds to several days.

Often the result will be between a few milliseconds and a few minutes.

If we record time in seconds, we lose precision which we want to avoid.

To do so we need to use the smallest possible measurement of time.

Since every milli-, micro-, or even nanosecond could be split in half for greater accuracy,

Less than an hour from now we will talk in more detail about continuous distributions

Now, regardless of whether we have a finite or infinite number of possibilities, we define

distributions using only two characteristics – mean and variance.

Simply put, the mean of the distribution is its average value.

Variance, on the other hand, is essentially how spread out the data is.

We measure this “spread” by how far away from the mean all the values are.

We denote the mean of a distribution as the Greek letter ‘mu’ and its variance as

When analysing distributions, it is important to understand what kind of data we have - population

Population data is the formal way of referring to “all” the data, while sample data is

For example, if an employer surveys an entire department about how they travel to work,

the data would represent the population of the department.

However, this same data would also be just a sample of the employees in the whole company.

Something to remember when using sample data is that we adopt different notation for the

We denote sample mean as “x bar” and sample variance as “s” squared.

One flaw of variance is that it is measured in squared units.

For example, if you are measuring time in seconds, the variance would be measured in

Usually, there is no direct interpretation of that value.

To make more sense of variance, we introduce a third characteristic of the distribution,

Standard deviation is simply the positive square root of variance.

As you can suspect, we denote it as “sigma” when dealing with a population, and as “s”

Unlike variance, standard deviation is measured in the same units as the mean.

Thus, we can directly interpret it and is often preferable.

One idea which we will use a lot is that any value between “mu minus sigma” and “mu

plus sigma” falls within one standard deviation away from the mean.

The more congested the middle of the distribution, the more data falls within that interval.

Similarly, the less data falls within the interval, the more dispersed the data is.

It is important to know there exists a constant relationship between mean and variance for

By definition, the variance equals the expected value of the squared difference from the mean

We denote this as “sigma squared, equals the expected value of Y minus mu, squared”.

After some simplification, this is equal to the expected value of “Y squared” minus

As we will see in the coming lectures, if we are dealing with a specific distribution,

Okay, when we are getting acquainted with a certain dataset we want to analyse or make

predictions with, we are most interested in the mean, variance and type of the distribution.

In our next video we will introduce several distributions and the characteristics they

In this lecture we are going to talk about various types of probability distributions

and what kind of events they can be used to describe.

Certain distributions share features, so we group them into types.

Some, like rolling a die or picking a card, have a finite number of outcomes.

They follow discrete distributions and we use the formulas we already introduced to

calculate their probabilities and expected values.

Others, like recording time and distance in track & field, have infinitely many outcomes.

They follow continuous distributions and we use different formulas from the once we mentioned

Throughout the course of this video we are going to examine the characteristics of some

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概率分佈簡介 (Introduction to Probability Distributions)

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