Normal distribution
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Normal Distribution. Introduction. Probability Density Functions. Probability Density Functions…. Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values.

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Normal Distribution

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Normal distribution

Normal Distribution

Introduction


Probability density functions

Probability Density Functions


Probability density functions1

Probability Density Functions…

  • Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values.

  •  We cannot list the possible values because there is an infinite number of them.

  •  Because there is an infinite number of values, the probability of each individual value is virtually 0.


Point probabilities are zero

Point Probabilities are Zero

  • Because there is an infinite number of values, the probability of each individual value is virtually 0.

    Thus, we can determine the probability of a range of values only.

  • E.g. with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say.

  • In a continuous setting (e.g. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0.

  • It is meaningful to talk about P(X ≤ 5).


Probability density function

Probability Density Function…

  • A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements:

  • f(x) ≥ 0 for all x between a and b, and

  • The total area under the curve between a and b is 1.0

f(x)

area=1

a

b

x


Uniform distribution

Uniform Distribution…

  • Consider the uniform probability distribution (sometimes called the rectangular probability distribution).

  • It is described by the function:

f(x)

a

b

x

area = width x height = (b – a) x = 1


Example

Example

  • The amount of petrol sold daily at a service station is uniformly distributed with a minimum of 2,000 litres and a maximum of 5,000 litres.

  • What is the probability that the service station will sell at least 4,000 litres?

  • Algebraically: what is P(X ≥ 4,000) ?

  • P(X ≥ 4,000) = (5,000 – 4,000) x (1/3000) = .3333

f(x)

2,000

5,000

x


Bin width 25

Bin width 25


Bin width 5

Bin width 5


Bin width 1

Bin width 1


Conditions for use of the normal distribution

Conditions for use of the Normal Distribution

  • The data must be continuous (or we can use a continuity correction to approximate the Normal)

  • The parameters must be established from a large number of trials


The normal distribution

The Normal Distribution…

  • The normal distribution is the most important of all probability distributions. The probability density function of a normal random variable is given by:

  • It looks like this:

  • Bell shaped,

  • Symmetrical around the mean …


The normal distribution1

The Normal Distribution…

  • Important things to note:

The normal distribution is fully defined by two parameters:

its standard deviation andmean

The normal distribution is bell shaped and

symmetrical about the mean

Unlike the range of the uniform distribution (a ≤ x ≤ b)

Normal distributions range from minus infinity to plus infinity


Standard normal distribution

0

1

1

Standard Normal Distribution…

  • A normal distribution whose mean is zero and standard deviation is one is called the standard normal distribution.

  • Any normal distribution can be converted to a standard normal distribution with simple algebra. This makes calculations much easier.


Normal distribution1

Normal Distribution…

  • Increasing the mean shifts the curve to the right…


Normal distribution2

Normal Distribution…

  • Increasing the standard deviation “flattens” the curve…


Calculating normal probabilities

Calculating Normal Probabilities…

  • Example: The time required to build a computer is normally distributed with a mean of 50 minutes and a standard deviation of 10 minutes:

  • What is the probability that a computer is assembled in a time between 45 and 60 minutes?

  • Algebraically speaking, what is P(45 < X < 60) ?

0


Calculating normal probabilities1

Calculating Normal Probabilities…

…mean of 50 minutes and a

standard deviation of 10 minutes…

  • P(45 < X < 60) ?

0


Distinguishing features

DistinguishingFeatures

  • The mean ± 1 standard deviation covers 66.7% of the area under the curve

  • The mean ± 2 standard deviation covers 95% of the area under the curve

  • The mean ± 3 standard deviation covers 99.7% of the area under the curve

Tripthi M. Mathew, MD, MPH


68 95 99 7 rule

68% of the data

95% of the data

99.7% of the data

68-95-99.7 Rule


Are my data normal

Are my data “normal”?

  • Not all continuous random variables are normally distributed!!

  • It is important to evaluate how well the data are approximated by a normal distribution


Are my data normally distributed

Are my data normally distributed?

  • Look at the histogram! Does it appear bell shaped?

  • Compute descriptive summary measures—are mean, median, and mode similar?

  • Do 2/3 of observations lie within 1 stddev of the mean? Do 95% of observations lie within 2 stddev of the mean?


Law of large numbers

Law of Large Numbers

  • Rest of course will be about using data statistics (x and s2) to estimate parameters of random variables ( and 2)

  • Law of Large Numbers: as the size of our data sample increases, the mean x of the observed data variable approaches the mean  of the population

  • If our sample is large enough, we can be confident that our sample mean is a good estimate of the population mean!

Stat 111 - Lecture 7 - Normal Distribution


Points of note

Points of note:

  • Total area = 1

  • Only have a probability from width

    • For an infinite number of z scores each point has a probability of 0 (for the single point)

  • Typically negative values are not reported

    • Symmetrical, therefore area below negative value = Area above its positive value

  • Always draw a sketch!


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