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# Normal Distribution - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about ' Normal Distribution' - quemby-huber

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Presentation Transcript

Introduction

### 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.

• 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).

• 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

• 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

• 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

• 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 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 …

• Important things to note:

The normal distribution is fully defined by two parameters:

its standard deviation andmean

The normal distribution is bell shaped and

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

Normal distributions range from minus infinity to plus infinity

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.

• Increasing the mean shifts the curve to the right…

• Increasing the standard deviation “flattens” the curve…

• 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

…mean of 50 minutes and a

standard deviation of 10 minutes…

• P(45 < X < 60) ?

0

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

95% of the data

99.7% of the data

68-95-99.7 Rule

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

• 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?

• 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

• 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!