# Continuous distributions

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STAM4000
Quantitative Methods
Week 4
Continuous distributions
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?
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 x
1, x2, ……….
For each value of X, we have a corresponding probability, p(x
1), 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
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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 σ
2).
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
~ 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 σ
2).
Common notation used to summarise shape/centre/spread for a normally distributed random variable
is
N