CONTINUOUS RANDOM VARIABLES AND THE NORMAL DISTRIBUTION
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CONTINUOUS RANDOM VARIABLES AND THE NORMAL
DISTRIBUTION
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CONTINUOUS PROBABILITY DISTRIBUTION
Two characteristics1. The probability that x assumes a value
in any interval lies in the range 0 to 12. The total probability of all the intervals
within which x can assume a value of 1.0
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Area under a curve between two points.
x = a x = b x
Shaded area is between 0 and 1
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Total area under a probability distribution curve.
Shaded area is 1.0 or 100%
x
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Area under the curve as probability.
a b x
Shaded area gives the probability P (a ≤ x ≤ b)
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The probability of a single value of x is zero.
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THE NORMAL DISTRIBUTION
Normal Probability Distribution A normal probability distribution ,
when plotted, gives a bell-shaped curve such that1. The total area under the curve is 1.0.2. The curve is symmetric about the mean.3. The two tails of the curve extend
indefinitely.
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Normal distribution with mean μ and standard deviation σ.
Standard deviation = σ
Mean = μ x
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Total area under a normal curve.
The shaded area is 1.0 or 100%
μ x
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A normal curve is symmetric about the mean.
Each of the two shaded areas is .5 or 50%
.5.5
μ x
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Areas of the normal curve beyond μ ± 3σ.
μ – 3σ μ + 3σ
Each of the two shaded areas is very close to zero
μ x
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Three normal distribution curves with the same mean but different standard deviations.
σ = 5
σ = 10
σ = 16
xμ = 50
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Three normal distribution curves with different means but the same standard deviation.
σ = 5 σ = 5 σ = 5
µ = 20 µ = 30 µ = 40 x
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THE STANDARD NORMAL DISTRIBTUION
Definition The normal distribution with μ = 0 and σ
= 1 is called the standard normal distribution.
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The standard normal distribution curve.
σ = 1
µ = 0
-3 -2 -1 0 1 2 3 z
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THE STANDARD NORMAL DISTRIBTUION
Definition The units marked on the horizontal axis
of the standard normal curve are denoted by z and are called the z values or z scores. A specific value of z gives the distance between the mean and the point represented by z in terms of the standard deviation.
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Area under the standard normal curve.
-3 -2 -1 0 1 2 3 z
. 5. 5
Each of these two areas is .5
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Examples Using The Standard Normal Table …
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STANDARDIZING A NORMAL DISTRIBUTION
Converting an x Value to a z Value For a normal random variable x, a particular
value of x can be converted to its corresponding z value by using the formula
where μ and σ are the mean and standard deviation of the normal distribution of x, respectively.
x
z
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Examples … IQ ~ Normal mean = 100 stdev = 15
…
…
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DETERMINING THE z AND x VALUES WHEN AN AREA UNDER THE NORMAL DISTRIBUTION CURVE IS KNOWN
Examples …
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Finding an x Value for a Normal Distribution
For a normal curve, with known values of μ and σ and for a given area under the curve the x value is calculated as
x = μ + zσ
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