1. Uniform and Normal Distributions
Chp. 6 Uniform and Normal
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2. Continuous Probability Distributions
Many continuous probability distributions, including:
Uniform
Normal
Gamma
Exponential
Chi-Squared
Lognormal
Weibull
Weibull PDF Source:
Uniform
Normal –
Gamma
Exponential
Chi-Squared
Lognormal
Weibull -
Chp. 6 Uniform and Normal
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3. Uniform Distribution
Simplest – characterized by the interval endpoints, A and B.
A ≤ x ≤ B
= 0 elsewhere
Mean and variance:
and
draw distribution
Chp. 6 Uniform and Normal
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4. Example: Uniform Distribution
A circuit board failure causes a shutdown of a computing system until a new board is delivered. The delivery time X is uniformly distributed between 1 and 5 days.
What is the probability that it will take 2 or more days for the circuit board to be delivered?
interval = [1,5]
f(x) = 1/(B-A) = 1/(5-1) = ¼, 1 < x < 5 (0 elsewhere)
Chp. 6 Uniform and Normal
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5. In Class Examples
6.3
Chp. 6 Uniform and Normal
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6. Normal Distribution The “bell-shaped curve”
Also called the Gaussian distribution
The most widely used distribution in statistical analysis
forms the basis for most of the parametric tests we’ll perform later in this course.
describes or approximates most phenomena in nature, industry, or research
Random variables (X) following this distribution are called normal random variables.
the parameters of the normal distribution are μ and σ (sometimes μ and σ2.)
note: nonparametric tests are distribution-free, assume no underlying distribution (see ch 16)
Chp. 6 Uniform and Normal
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7. Normal Distribution
The density function of the normal random variable X, with mean μ and variance σ2, is
all x.
(μ = 5, σ = 1.5)
properties of the curve:
peak is both the mean and the mode and the median and occurs at x = μ
curve is symmetrical about a vertical axis through the mean
total area under the curve and above the horizontal axis = 1.
Chp. 6 Uniform and Normal
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8. Standard Normal RV …
Note: the probability of X taking on any value between x1 and x2 is given by:
To ease calculations, we define a normal random variable
where Z is normally distributed with μ = 0 and σ2 = 1
We cannot integrate this expression simply, so we will use the Z transformation.
Chp. 6 Uniform and Normal
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9. Standard Normal Distribution
Table A.3 Pages : “Areas under the Normal Curve”
Body of table – area under curve
Chp. 6 Uniform and Normal
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10. Examples P(Z ≤ 1) = P(Z ≥ -1) = P(-0.45 ≤ Z ≤ 0.36) =
draw the area on the picture …
2012
1. P(Z < 1) =
2. P(Z ≥ -1) =
3. P(-0.45 ≤ Z ≤ 0.36) = P(Z < 0.36) – P(Z< -0.45) = – =
Chp. 6 Uniform and Normal
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11. Name:________________________
Use Table A.3 to determine (draw the picture!)
1. P(Z ≤ 0.8) =
2. P(Z ≥ 1.96) =
3. P(-0.25 ≤ Z ≤ 0.15) =
4. P(Z ≤ -2.0 or Z ≥ 2.0) =
1. P(Z ≤ 0.8) =
2. P(Z ≥ 1.96) = 1 – = (=P(Z < -1.96)) Note symmetry!!
3. P(-0.25 ≤ Z ≤ 0.15) = – =
4. P(Z ≤ -2.0 or Z ≥ 2.0) = 2 * =
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12. Applications of the Normal Distribution
A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance less than 44 ohms?
Solution: X is normally distributed with μ = 40 and σ = 2 and x = 44
P(X<44) = P(Z< +2.0) =
Therefore, we conclude that 97.72% will have a resistance less than 44 ohms What percentage will have a resistance greater than 44 ohms?
Chp. 6 Uniform and Normal
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13. The Normal Distribution “In Reverse”
Example:
Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X.
Step 1
If P(Z < z) = 0.45,
z = _______ (from Table A.3)
Why is z negative?
Step 2
X = _________
45%
Will X be greater than 40?
k =
Z = = (X – 40)/6
X = (-0.125*6)+40 = 39.25
Chp. 6 Uniform and Normal
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14. In-Class Exercise 6.14
The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.
What proportion of rings will have inside diameters exceeding centimeters?
(b) What is the probability that a piston ring will have an inside diameter between 9.97 and centimeters?
(c) Below what value of inside diameter will 15% of the piston rings fall? of 0.03 centimeter.
Chp. 6 Uniform and Normal
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15. In-Class Exercise 6.14 Solution
What proportion of rings will have inside diameters exceeding centimeters?
(b) What is the probability that a piston ring will have an inside diameter between 9.97 and centimeters?
(c) Below what value of inside diameter will 15% of the piston rings fall? of 0.03 centimeter.
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16. In-Class Exercise 6.17
The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered? Assume that the lifetime of a motor follows a normal distribution.
Chp. 6 Uniform and Normal
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17. In-Class Exercise 6.17
The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered? Assume that the lifetime of a motor follows a normal distribution. 3%
Solve for X
X = (2 * -1.88) + 10 = 6.24
A z-value of corresponds to 3% of area under the curve
Chp. 6 Uniform and Normal
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