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