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STAM4000

Quantitative Methods

Week 4

Continuous distributions

https://www.reddit.com/r/sciencememes/comments/kdvn1y/a_basic_understanding_of_statistics_is_required/

Frequency curve, is a smoothed

histogram.

Quantitative variable on the

horizontal (X axis).

Frequency or relative frequency on

the vertical (or y axis)

Joke: here the “PEAK” is the “modal

class” and it has the highest

frequency BUT the highest value on

the horizontal axis is to the RIGHT of

the peak

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3 #1 #2 #3 Compare continuous random variables, continuous QUANTITATIVE or numerical variable, numbers with units. Solve reverse normal problems follow a different set of steps. Find probabilities with Z tables: Statistical tables called Z tables or Z values. FOLLOW STEPS |
Week 4 Continuous distributions Learning Outcomes |

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Why does

this

matter?

https://towardsdatascience.com/all-you-need-to-know-about-normal-distribution-3f67df0691f8

Why does this matter?

There are lots of types of probability distributions.

For example, the normal distribution is the most important probability distribution in statistics

because it fits many natural phenomena like heights, IQ etc. and because of the Central Limit Theorem

(next class). The normal distribution is also known as the Gaussian distribution, after Johann Carl

Friedrich Gauss 1777 – 1855. The graph of a normal distribution is unimodal and symmetric and may

be described as a normal curve, bell curve or mound curve.

Another type of probability distribution is the exponential distribution – ever present in the media

from 2020 due to COVID-19. The exponential distribution describes accelerated growth or decline.

For example, the exponential distribution can be used to model repose time periods (the length of the

time intervals between deaths from COVID-19) and for predicting probabilities of new deaths that

occur within pre-determined time intervals. (Source:

https://jptcp.com/index.php/jptcp/article/view/721/711)

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#1 Compare continuous random variables

https://www.pinterest.com.au/pin/37647346857491672/

Comparing

CONTINUOUS

random variables

that have

DIFFERENT units of

measure BY

STANDARDIZING or

TRANSFORMING

the random

variable to a Z

SCORE or Z value

or a

STANDARDIZED

SCORE; The latter

are UNITS FREE.

What is probability?

In general, probability is the likelihood that something will happen.

Most events are uncertain: probability theory is a way of quantifying the uncertainty.

In any case, a probability is a number which must obey certain rules.

The most basic rules of probability:

• 0 ≤ P(x) ≤ 1

• ∑ P(x) = 1

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Random Variables are

variables whose value

vary due to chance.

Think of these values as

possible outcomes of an

experiment.

Since these values are

random, we can assign a

probability to them.

WEEK 3:

Discrete

quantitative

random

variables can

take a distinct

value in a

range of

values.

E.g., marks in a

test

5, 5.5, 6, 6.5

WEEK 4:

Continuous

quantitative

random

variables can

take any value

within a range

of values.

E.g., anything

measurable like

height, weight,

sales, share price

#1 Compare continuous random variables

Recall, random variables …

Probability distributions

Say X is a random variable, that can take values x1, x2, ……….

For each value of X, we have a corresponding probability, p(x1), p(x2), ………

For a discrete random variable, there is a finite (limited) number of values X that can

take with the corresponding probabilities of X.

For a continuous random variable, there is an infinite (unlimited) number of values

that X can take with the corresponding probabilities of X.

This week, we will concentrate on continuous probability distributions.

The normal distribution of a variable, X, is unimodal and symmetric with mean μ and

standard deviation σi Each of these are in the same units as the original data.

The standard normal distribution, Z, is unimodal and symmetric with mean 0 and

standard deviation 1. Each of these are units free.

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Probability distributions or probability collection

A probability distribution is

a collection of possible

values, with the

corresponding

probabilities of each value.

A function that represents

a continuous

probability distribution is

called a probability density

function.

This week,

we will

concentrate

on

continuous

probability

distributions

Normal distribution of a

variable, X

•Standard normal

distribution, Z

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• The term “density” is used when describing

the rule to obtain probability information

for a continuous random variable.

• The normal distribution density function

has equation (the equation of the curve)

Normal distribution density function

f(x)

µ X

#1

where,

µ = mean of X variable

σ = standard deviation of X variable

π = 3.1415

e = 2.71828

x = value of X variable

f(x) = density of X value

The normal curve is asymptotic as it

never touches the horizontal axis and

continues in the left and right tail

to -∞ and +∞,respectively .

Normal distribution density function

For a continuous random variable, X, whose distribution is unimodal and symmetric, we say that X

is normally distributed, with mean μ and standard deviation σ, (the variance is σ2).

We will not be using the formula for the actual curve – instead we will be using other formulae to

“transform” values of a continuous normally distributed random variable X into another

continuous random variable – more later.

Why study the normal distribution?

• The normal distribution is the most famous continuous probability distribution (model).

• It is the most widely used distribution in statistics.

• Many measurable quantities in everyday life follow normal distributions. Example : IQ test

scores, weight, height etc.

• We need normality when estimating the population mean and the population proportion with

confidence intervals (see later in course).

• We need normality when testing hypotheses about the mean (see later in course).

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Week 2:

μ = population mean, read as “mu

σ = population standard deviation, read as “sigma”

x is a value of our continuous random variable, X

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• The curve is unimodal and symmetric

• The normal distribution density looks like a

“bell shaped frequency curve”.

• X is a continuous random variable.

• Here, X is normally distributed

• µ = population mean of X

• σ = population standard deviation of X

• Common notation used to summarize

shape/centre/spread: X ~ N(µ, σ)

More on the normal distribution

f(x)

µ X

= Median, 50th percentile

= Mode

#1

Total area beneath the curve has

a probability of 1 or 100%

What is the P(X < μ) = 0.50

50% = 0.50 |
50% = 0.50 |

Normal distribution density function

For a continuous random variable, X, whose distribution is unimodal and symmetric, we say that X is

normally distributed, with mean μ and standard deviation σ, (the variance is σ2).

Common notation used to summarise shape/centre/spread for a normally distributed random variable

is N