1. Continuous Probability Distributions
Recall:
P(a < X < b) = = F(b) – F(a)
F (a) =
μ = E[X] =
s2 = E[X2] – μ2
F(x)
f(x) x x
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2. Continuous Uniform Distribution
f(x;A,B) = 1/(B-A), A < x < B
where [A,B] is an interval on the real number line
μ = (A + B)/2 and s2 = (B-A)^2/12
1/(B-A)
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3. Continuous Uniform Distribution
EX: Let X be the uniform continuous random variable that denotes the current measured in a copper wire in milliamperes. 0 < x < 50 mA. The probability density function of X is uniform.
f(x) = , 0 < x < 50
What is the probability that a measurement of current is between 20 and 30 mA?
What is the expected current passing through the wire?
What is the variance of the current passing through the wire?
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4. Normal Distribution
n(x;μ,σ) =
E[X] = μ and Var[X] = s2
f(x) x
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5. Normal Distribution
Many phenomena in nature, industry and research follow this bell-shaped distribution.
Physical measurements
Rainfall studies
Measurement error
There are an infinite number of normal distributions, each with a specified μ and s.
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6. Normal Distribution Characteristics Bell-shaped curve
- < x < +
μ determines distribution location and is the highest point on curve
Curve is symmetric about μ
s determines distribution spread
Curve has its points of inflection at μ + s
μ + 1s covers 68% of the distribution
μ + 2s covers 95% of the distribution
μ + 3s covers 99.7% of the distribution
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7. Normal Distribution s s s s μ
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8. Normal Distribution n(x; μ = 0, s = 1) n(x; μ = 5, s = 1) f(x) x
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9. Normal Distribution n(x; μ = 0, s = 0.5) f(x) n(x; μ = 0, s = 1) x
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10. Normal Distribution n(x; μ = 5, s = .5) f(x) n(x; μ = 0, s = 1) x
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11. Normal Distribution μ + 1s covers 68% μ + 2s covers 95%
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12. Standard Normal Distribution
The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution.
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13. Standard Normal Distribution
The letter Z is traditionally used to represent a standard normal random variable.
z is used to represent a particular value of Z.
The standard normal distribution has been tabularized.
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14. Standard Normal Distribution
Given a standard normal distribution, find the area under the curve
(a) to the left of z = -1.85
(b) to the left of z = 2.01
(c) to the right of z = –0.99
(d) to right of z = 1.50
(e) between z = and z = 0.58
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15. Standard Normal Distribution
Given a standard normal distribution, find the value of k such that
(a) P(Z < k) = .1271
(b) P(Z < k) = .9495
(c) P(Z > k) = .8186
(d) P(Z > k) = .0073
(e) P( 0.90 < Z < k) = .1806
(f) P( k < Z < 1.02) = .1464
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16. Normal Distribution
Any normal random variable, X, can be converted to a standard normal random variable:
z = (x – μx)/sx
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17. Normal Distribution
Given a random Variable X having a normal distribution with μx = 10 and sx = 2, find the probability that X < 8. z x
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18. Normal Distribution
EX: The engineer responsible for a line that produces ball bearings knows that the diameter of the ball bearings follows a normal distribution with a mean of 10 mm and a standard deviation of 0.5 mm. If an assembly using the ball bearings, requires ball bearings mm, what percentage of the ball bearings can the engineer expect to be able to use?
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19. Normal Distribution EX: Same line of ball bearings.
What is the probability that a randomly chosen ball bearing will have a diameter less than 9.75 mm?
What percent of ball bearings can be expected to have a diameter greater than 9.75 mm?
What is the expected diameter for the ball bearings?
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20. Normal Distribution
EX: If a certain light bulb has a life that is normally distributed with a mean of 1000 hours and a standard deviation of 50 hours, what lifetime should be placed in a guarantee so that we can expect only 5% of the light bulbs to be subject to claim?
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21. Normal Distribution
EX: A filling machine produces “16oz” bottles of Pepsi whose fill are normally distributed with a standard deviation of 0.25 oz. At what nominal (mean) fill should the machine be set so that no more than 5% of the bottles produced have a fill less than oz?
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22. Relationship between the Normal and Binomial Distributions
The normal distribution is often a good approximation to a discrete distribution when the discrete distribution takes on a symmetric bell shape.
Some distributions converge to the normal as their parameters approach certain limits.
Theorem 6.2: If X is a binomial random variable with mean μ = np and variance s2 = npq, then the limiting form of the distribution of Z = (X – np)/(npq).5 as n  , is the standard normal distribution, n(z;0,1).
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23. Relationship between the Normal and Binomial Distributions
Consider b(x;15,0.4). Bars are calculated from binomial. Curve is normal approximation to binomial.
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24. Relationship between the Normal and Binomial Distributions
Let X be a binomial random variable with n=15 and p=0.4.
P(X=5) = ?
Using normal approximation to binomial:
n(3.5<x<4.5;μ=6, s=1.897) = ?
Note, μ = np = (15)(0.4) = 6
s = (npq).5 = ((15)(0.4)(0.6)).5 = 1.897
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25. Relationship between the Normal and Binomial Distributions
EX: Suppose 45% of all drivers in a certain state regularly wear seat belts. A random sample of 100 drivers is selected. What is the probability that at least 65 of the drivers in the sample regularly wear a seatbelt?
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