Continuous Random Variables and the Normal Distribution Dr Tom Ilvento
Department of Food and Resource Economics
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Most intro stat class would have a section on probability - we don’t
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But it is important to get exposure to the normal distribution
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We will use this distribution, and the related tdistribution, when we shift to inferences
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First we need to understand the normal distribution And feel comfortable with the Standard Normal Table 2
Probability
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Random Variables
Probability is a numerical measure of the likelihood that Event A will occur
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P(A)
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Random Variables – variables that assume numerical values associated with random outcomes from an experiment
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Random variables can be:
Prob(A)
The basic definition is:
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It is a proportion which goes from 0 to 1
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Discrete Continuous
For random variables there is
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A probability distribution Expectation and variance 4
The probability of the number of males in three live births
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This discrete distribution shows the probability of 0, 1, 2, or 3 males in three births The mean and variance are:
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Mean = 1.5 Variance = .75 Std Dev = .866
What are Continuous Random Variables?
Probability Distribution p(X) 0.4 0.3 0.2 0.1 0 0
1
2
3
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Unlike Discrete Random Variables, Continuous Random Variables take on any point in the interval
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Thus the probability distribution is continuous It is referred to as a Probability Density Function
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PDF f(x)
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You might want to get a copy of the Standard Normal Distribution handout
When dealing with a pdf...
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It is not particularly useful to think of a probability when a continuous random variable takes on a particular value
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P(x=a) = 0
But, we can think of areas under the curve as reflecting a probability
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P(a 1.28 z > 1.645 z > 2.33
Two Tailed z < -1.645 or z > 1.645 z < -1.96 or z > 1.96 z < -2.575 or z > 2.575
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Infinate Number of Normal Curves
The Normal Distribution
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One bell shaped, symmetrical distribution is the normal distribution It is defined by two parameters
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µ the Mean ! The Standard Deviation
f ( x) =
2 1 e #(1/ 2 )[( x # µ ) / ! ] ! 2"
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For every distribution with a mean (µ) and a standard deviation (!) there is a different normal curve
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Thus, there are an infinite number of normal curves If x is a random variable distributed as a normal variable then it is designated as: x ~ N(mean, std dev)
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Properties of the Normal Distribution
Standard Normal Distribution
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The area under the curve = 1
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Mean = Median = Mode
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It has an infinite range
The IQR is 1.33 Std Deviations wide (.677 below or .677 above)
z x F(x)
0.025 0.02
0 100 50.00%
50
1.1 1 0.9 0.8 0.7
0.015
0.6 0.5
50.00%
0.01
0.4 0.3
0.005
0.2
Cumulative Probability
Defined by the mean and standard deviation
Cumulative Probability Graph
Probability Density
Symmetrical, Bell-shaped curve
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0.1 0 0
50
100
150
0 200
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Since its properties are defined by a formula, we can a priori define probabilities associated with the normal curve, but each combination of a mean and std deviation results in a different normal curve If we convert our normally distributed variable to z-scores, we make it possible to use one table of probabilities for all normal pdf
Cumulative Probability Graph z x F(x)
0 0 50.00%
0.4550
1.1 1
0.4
0.9
0.35
0.8
0.3
0.7
0.25
0.6
0.2
0.5
50.00%
0.4
0.15
0.3
0.1
0.2
0.05
0.1
0 -6
-4
-2
0 0
2
4
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This is called the Standard Normal Distribution
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mean = 0 std dev = 1
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Cumulative Probability
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For every variable distributed normally with a mean (µ) and a standard deviation (!) there is a different normal curve
Probability Density
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Finding Areas under the Normal curve
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Look at the Standard Normal Table
Basic Steps 1. Draw the curve and the area we are interested in 2. Convert the values to zscores
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There are several types of tables
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Since the distribution is symmetrical,
We will work with a table where only ! of the curve is presented
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3. Read the proportions in the table, and do any additional calculations that may be necessary
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Both halves are identical, and each half represents p = .5
So our table will only calculate probabilities for the right hand side of the distribution
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Moving from the center, µ = 0, toward the right tail
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Standard Normal Table - a partial view of the table •
The table allows for two decimal places of a zscore
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Vertical axis is the ones and first decimal place
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Horizontal axis is the second decimal place
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To find the probability associated with a zscore of 1.08
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Z
This value represents the probability from µ = 0 up to 1.08 standard deviations above the mean - .3599
