Original subtitles

Instructor: Welcome back, in this lecture,

we are going to focus on the continuous

logistic probability distribution.

We denote a logistic distribution

with the entire word logistic, followed by two parameters.

It's mean and scale parameter,

like the one for the exponential distribution.

We also refer to the mean parameter as the location

and we shall use the terms interchangeably

for the remainder of the video.

Thus, we read the statement below as:

variable Y follows a logistic distribution

with location 6 and a scale of 3.

All right, we often encounter logistic distributions

when trying to determine how continuous variable inputs

can affect the probability of a binary outcome.

This approach is commonly found

in forecasting competitive sports events where there exist

only two clear outcomes, victory or defeat.

For instance, we can analyze whether the average speed

of a tennis player's serve plays

a crucial role in the outcome of the match.

Expectation dictates that sending the ball

with higher velocity leaves opponents

with a shorter period to respond.

This usually results in a better hit,

which could lead to a point for the server.

To reach the highest speeds, tennis players often give up

some control over the shot, so are less accurate.

Therefore, we cannot assume that there is

a linear relationship between point conversion

and serve speeds.

Theory suggests there exists some optimal speed

which enables the serve to still be accurate enough.

Then, most of the shots we convert into points

will likely have similar velocities.

As tennis players go further away from the optimal speed

their shots either become too slow

and easy to handle, or too inaccurate.

This suggests that the graph of the PDF

of the logistic distribution would look similarly

to the normal distribution.

Actually, the graph of the logistic distribution

is defined by two key features:

its mean and its scale parameter.

The former dictates the center of the graph.

Whilst the latter shows how spread out

the graph is going to be.

Going back to the tennis example,

the mean would represent the optimal speed

whilst the scale would dictate

how lenient we can be with the hit.

To elaborate, some tennis players

can hit a great serve further away

from the optimal speed than others.

For instance, Serena Williams can hit fantastic serves

even if the ball moves faster,

or slower than it optimally should.

Therefore, she is going to have a more spread

out PDF than some of her opponents.

Fantastic now, let's discuss

the cumulative distribution function.

It should be a curve that starts off slow,

then picks up rather quickly

before plateauing around the 1 mark.

That is because once we reach values near the mean

the probability of converting the point drastically goes up.

Once again, the scale would dictate the shape of the graph.

In this case, the smaller the scale

the later the graph starts to pick up,

but the quicker it reaches values close to 1.

Okay, you can use expected values to estimate

the variance of the distribution.

To avoid confusing mathematical expressions

you only need to know it is equal

to the square of the scale times pie squared over 3.

Great job everybody, now that you know

all these various types of distributions

we can explore how probability features in other fields.

In the next section of this course,

we are going to focus on statistics,

data science and other related fields,

which integrate probability, thanks for watching.