Continuous probability distributions Examples The probability that the
Continuous probability distributions Examples: The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is _____ The probability that a given part will fail before 1000 hours of use is _____ In general, 1 _____ ETM 620 - 09 U
Understanding continuous distributions The probability that the average daily temperature in Georgia during the month of August falls between 90 and 95 degrees is The probability that a given part will fail before 1000 hours of use is 2 ETM 620 - 09 U
Continuous probability distributions The continuous probability density function (pdf) f(x) ≥ 0, for all x ∈ R The cumulative distribution, F(x) Expectation and variance 3 μ = E(X) = _______ σ2 = ________ ETM 620 - 09 U
An example f(x) = { x, 2 -x, 0, 0<x<1 1≤x<2 elsewhere 1 st – what does the function look like? a) P(X < 1. 2) = __________ b) P(0. 5 < X < 1) = __________ c) E(X) = -∞∫∞ x f(x) dx = ____________ 4 ETM 620 - 09 U
Continuous probability distributions Many continuous probability distributions, including: Uniform Exponential Gamma Weibull Normal Lognormal Bivariate normal 5 ETM 620 - 09 U
Uniform distribution Simplest – characterized by the interval endpoints, A and B. A≤x≤B =0 elsewhere Mean and variance: and 6 ETM 620 - 09 U
Example 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? 7 ETM 620 - 09 U
Gamma & Exponential distributions Recall the Poisson Process Number of occurrences in a given interval or region “Memoryless” process Sometimes we’re interested in the time or area until a certain number of events occur. For example An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to a minute will elapse before 2 calls arrive? How long before the next call? 8 ETM 620 - 09 U
Gamma distribution The density function of the random variable X with gamma distribution having parameters α (number of occurrences) and β (time or region, 1/λ). x > 0. 9 ETM 620 - 09 U
Exponential distribution Special case of the gamma distribution with α = 1. x > 0. Describes the time until or time between Poisson events. μ=β σ 2 = β 2 10 ETM 620 - 09 U
Example An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to a minute will elapse before 2 calls arrive? β = ____ α = ____ P(X ≤ 1) = _________________ 11 ETM 620 - 09 U
Example (cont. ) What is the expected time before the next call arrives? β = ____ α = ____ μ = _________________ 12 ETM 620 - 09 U
Another example Example 6 -6, page 136. Note: in Excel, use 13 =GAMMADIST(x, alpha, beta, cumulative) to find the probability associated with a specific value of x. Use =GAMMAINV(probability, alpha, beta) to find the value of ETM 620 - 09 U x given a probability.
Wiebull Distribution Used for many of the same applications as the gamma and exponential distributions, but does not require memoryless property of the exponential 14 ETM 620 - 09 U
Example Designers of wind turbines for power generation are interested in accurately describing variations in wind speed, which in a certain location can be described using the Weibull distribution with α = 0. 02 and β = 2. A designer is interested in determining the probability that the wind speed in that location is between 3 and 7 mph. P(3 < X < 7) = ______________ In Excel, =WEIBULL(x, alpha, beta, cumulative) 15 ETM 620 - 09 U
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. ) 16 ETM 620 - 09 U
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) In Excel, =NORMDIST(x, mean, standard_dev, cumulative) 17 and =NORMINV(probability, mean, standard_dev) ETM 620 - 09 U
Standard normal RV … Note: the probability of X taking on any value between x 1 and x 2 is given by: To ease calculations, we define a normal random variable where Z is normally distributed with μ = 0 and σ2 = 1 and, 18 ETM 620 - 09 U
Standard normal distribution Table II, pg. 601 -602: “Cumulative Standard Normal Distribution” In Excel, =NORMSDIST(z) and =NORMSINV(probability) 19 ETM 620 - 09 U
Examples P(Z ≤ 1) = Φ(1) = P(Z ≥ -1) = P(-0. 45 ≤ Z ≤ 0. 36) = 20 ETM 620 - 09 U
Your turn … Use Excel 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) = 21 ETM 620 - 09 U
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. 1) If Φ(k) = 0. 45, k = ______ 2) Z = _______ X = _____ 22 ETM 620 - 09 U
Normal approximation to the binomial If n is large and p is not close to 0 or 1, or if n is smaller but p is close to 0. 5, then the binomial distribution can be approximated by the normal distribution using the transformation: NOTE: add or subtract 0. 5 from X to be sure the value of interest is included (draw a picture to know which) Look at example 7 -10, pg. 156 23 ETM 620 - 09 U
Another example The probability of a patient making a full recovery from a rare heart ailment is 0. 4. If 100 people are diagnosed with this ailment, what is the probability that fewer than 30 of them recover? p = 0. 4 n = 100 μ = ______ σ = _______ if x = 30, then z = ___________ and, P(X < 30) = P (Z < _____) = _____ 24 ETM 620 - 09 U
Your Turn DRAW THE PICTURE!! Refer to the previous example, 25 1. What is the probability that more than 50 survive? 2. What is the probability that exactly 45 survive? ETM 620 - 09 U
Lognormal Distribution • When the random variable Y = ln(X) is normally distributed with mean μ and standard deviation σ, then X has a lognormal distribution with the density function, 26 ETM 620 - 09 U
Example The average rate of water usage (in thousands of gallons per hour) by a certain community is know to involve the lognormal distribution with parameters μ = 5 and σ =2. What is the probability that, for any given hour, more than 50, 000 gallons are used? P(X > 50, 000) = _____________ 27 ETM 620 - 09 U
Bivariate Normal distribution Random variables [X, Y] that follow a bivariate normal distribution have a joint density function, and a joint probability, This would be pretty nasty to calculate, except … X ∼ N (μX, σ2 X ) 28 and Y ∼ N (μY , σ2 Y ) ETM 620 - 09 U
So… It can be shown (see pp. 161 -162) that, This allows us to use the standard normal approach to solve several important problems. 29 ETM 620 - 09 U
Example 7 -14, pg. 163 Given the information regarding shear strength and weld diameter in the example, what is the probability that the strength specification (1080 lbs. ) can be met if the weld diameter is 0. 18 in. ? 30 ETM 620 - 09 U
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