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Random Variables and Probability Distributions KANCHALA SUDTACHAT Content 1. Random Variables 9. Normal Distribution 2. Probability 10. t- Distribution 3. Discrete Random Variables 11. Exponential Distribution 4. Discrete Uniform Distribution 5. Binomial Distribution 12. Normal Approximation to the Binomial and Poisson Distributions 6. Poisson Distribution 13. Normal Probability Plots 7. Continuous Random Variables 8. Continuous Uniform Distribution Random Experiment Random Variables A measurement is usually denoted by a variable such as X. A variable whose measured value can change. (random variable) Discrete Random Variable – a finite set of real numbers Continuous Random Variable – an interval of real numbers Probability A random variable is used to describe a measurement. Probability is used to quantify the likelihood, that a measurement falls within some set of values. “The chance that X, the length of a manufactured part, is between 10.8 and 11.2 millimeters is 25%” Probability Degree of belief A relative frequency (or proportion) of repeated replicates that fall in the interval will be percent, uses a long run proportion. Probability Properties Given a set E, the set of real number is denoted as R. Each element is in one and only one of the sets E1, E2,…,Ek. Applying Probability Properties Ex: X denotes the life in hours of standard fluorescent tubes P(X ≤ 5000) = 0.1, P(5000<X ≤ 6000) = 0.3, P(X > 8000) = 0.4 Discrete Random Variables Example 3-19 A voice communication network for a business contains 48 external lines. At a particular time, the system is observed and some of the lines are being used. Example 3-20 The analysis of the surface of a semiconductor wafer records the number of particles of contamination that exceed a certain size. Define the random variable X to equal the number of particles of contamination. Probability Mass Function (pmf) The probability distribution of a random variable X is a description of the probabilities associated with the possible values of X. It is convenient to express the probability in terms of a formula. Example 3-21 There is a chance that a bit transmitted through a digital transmission channel is received in error. Let X equal the number of bits in error in the next 4 bits transmitted. The possible value for X are {0, 1, 2, 3, 4} Probability Mass Function (pmf) Cumulative Distribution Function (cdf) Cumulative Distribution Function (cdf) Example 3-22 In previous example, the probability mass function for X P(X=0) = 0.6561 P(X=1) = 0.2916 P(X=3) = 0.0036 P(X=4) = 0.0001 P(X=2) = 0.0486 Mean and Variance Example 3-23 For the number of bits in error in the previous example. Determining µ and σ2 Mean and Variance Example 3-24 Two new product designs are to be compared on the basis for revenue potential. Marketing feels that the revenue from design A can be predicted quite accurately to be $3 million. The revenue potential of design B is more difficult to assess. Marketing concludes that there is a probability of 0.3 that the revenue from design B will be $7 million, but there is a 0.7 probability that the revenue will be only $2 million. Which design would you choose? Discrete Uniform Distribution Example The first digit of a part’s serial number is equally likely to be any one of the digits 0 through 9. If one part is selected from a large batch and X is the first digit of the serial number. Discrete Uniform Distribution Example As in previous example 3-19, let the random variable X denote the number of the 48 voice lines that are in use at a particular time. Assume that X is a discrete uniform random variable with a range of 0 to 48. Determining µ and σ2 Binomial Distribution Each of these random experiments can be thought of as consisting of a series of repeated, random trials. The random variable in each case is a count of the number of trials that meet a specified criterion. Binomial Distribution Binomial Distribution Example 3-25 In Example 3-21, assume that the chance that a bit transmitted through a digital transmission channel is received in error is 0.1. Also assume that the transmission trials are independent. Let X = the number of bits in error in the next 4 bits transmitted. Determine P (X=2) Binomial Distribution Example 3-27 Each sample of water has a 10% chance of containing high levels of organic solids. Assume that the samples are independent with regard to the presence of the solids. Determine the probability that in the next 18 samples, exactly 2 contain high solids. Example 3-28 For the number of transmitted bits received in error in Example 3-21, n = 4 and p = 0.1 Poisson Process Events occur randomly in an interval time. The number of events over an interval time is a discrete random variable. Poisson Process Example 3-30 Flaws occur at random along the length of the thin copper wire. Let X denote the random variable that counts the number of flaws in a length of L millimeters of wire and suppose that the average number of flaws in L millimeters is λ. Poisson Process In general, consider an interval T of real number partitioned into subintervals of small length Δt and assume that as Δt tends to zero, 1) The probability of more than one event in a subinterval tends to zero, 2) The probability of one event in a subinterval tends to λ Δt /T 3) The event in each subinterval is independent of other subintervals A random experiment with these properties is called a Poisson Process. Poisson Process Poisson Process Example 3-31 For the case of the thin copper wire, suppose that the number of flaws follows a Poisson distribution with a mean of 2.3 flaws per millimeter. Determine the probability of exactly 2 flaws in 1 millimeter of wire. Poisson Process Example 3-32 Contamination is a problem in the manufacture of optical storage disks. The number of particles of contamination that occur on an optical disk has a Poisson distribution, and the average number of particles per centimeter squared of media surface is 0.1. The area of a disk under study is 100 squared centimeters. Determine the probability that 12 particles occur in the area of a disk under study. Continuous Random Variables Probability density Function f(x) are used to describe physical systems. Considering the density of a loading on a long, thin beam. For any point x along the beam, the density can be described by a functions (in grams/cm). Probability density function A pdf is zero for x values that cannot occur f(x) is used to calculate an area the represents the probability that X assumes a value in [a, b]. Probability density function Example 3-2 Let the continuous random variable X denote the current measured in a thin copper wire in mill-amperes. Assume that the range of X is [0, 20 mA], and assume that the probability density function of X is f(x)=0.05 for 0 ≤ x ≤ 20. What is the probability that a current measurement is less than 10 mill-amperes? Probability density function Example 3-3 Let the continuous random variable X denote the distance in micrometers from the start of a track on a magnetic disk until the first flaw. Historical data show that the distribution of X can be modeled by a pdf For what proportion of disks is the distance to the first flaw greater than 1000 micrometers? Cumulative Distribution Function Cumulative Distribution Function Example 3-4 Consider the distance to flaws in Example 3-3 with pdf f(x) = 1/2000 exp(-x/2000) For x ≥ 0. Determine the cdf. Mean and Variance Mean and Variance Example 3-5 For the copper current measurement in Example 3-2, the mean of X is Example 3-6 of X is For the distance to a flaw in Example 3-3, the mean Normal Distribution Histograms have characteristic shape as bell shapes. The random variable that equals the average result over the replicates tends to have a normal distribution as the number of replicates becomes large. Normal Distribution Normal Distribution Example 3-7 Assume that the current measurements in a strip of wire follow a normal distribution with a mean of 10 mill-amperes and a variance of 4 mill-amperes2. What is the probability that a measurement exceeds 13 mill-amperes? Normal Distribution Standard normal random variable Using standard normal random variable Example 3-8 Assume that Z is a standard normal random variable. Appendix A Table I provides probabilities of the form P(Z ≤ z). The use of Table I to fine P(Z ≤ 1.5). Normal Distribution 39 Example 3-9 1) P(Z > 1.26) 2) P(Z < -0.86) 3) P(Z > -1.37) 4) P(Z < 1.37) 5) P(-1.25 < Z < 0.37) 6) P(Z ≤ -4.6) 7) Fine the value z such that P(Z > z) = 0.05 8) Find the value of z such that P(-z < Z < z)=0.99 Example 3-10 Suppose the current measurements in a strip of wire follow a normal distribution with a mean of 10 mill-amperes and a variance of 4 millamperes2. What is the probability that a measurement exceeds 13 millamperes? Standardizing Example 3-11 Continuing the previous example, what is the probability that a current measurement is between 9 and 11 mill-amperes? Example 3-12 In the transmission of a digital signal, assume that the background noise follows a normal distribution with a mean of 0 volt and standard deviation of 0.45 volt. If the system assumes that a digital 1 has been transmitted when the voltage exceeds 0.9, what is the probability of detecting a digital 1 when none was sent? Example 3-13 The diameter of a shaft in storage drive is normally distributed with mean 0.2508 inch and standard deviation 0.0005 inch. The specifications on the shaft are 0.2500 ± 0.0015 inch. What proportion of shafts conforms to specifications? t-Distribution When is unknown Small sample size Degree of freedom (k) = n-1 Significant level = t, k 46 t-Distribution 47 Exponential Distribution Let the random variable X denote the length from any starting point on the wire until a flaw is detected. Let the random variable N denote the number of flaws in x mill-meters of wire. Assume that the mean number of flaws is λ per mill-meters. Exponential Distribution Example 3-33 In a large corporate computer network, use log-on to the system can be modeled as a Poisson process with a mean of 25 log-on per hour. What is the probability that there are no log-on in an interval of 6 minutes? Normal Approximation to the Binomial A random variable can be approximated with a normal random variable when n is large. Normal Approximation to the Binomial Example 3-34 In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable, and assume that the probability that a bit is received in error in 1x10-5. If 16 million bits are transmitted, what is the probability that more than 150 errors occur? Example 3-35 Again consider the transmission of bits in the previous example. To judge how well the normal approximation works, assume that only n=50 bits are to be transmitted and that the probability of an error is p=0.1. The exact probability that 2 or fewer errors occur is Normal Approximation to the Poisson Recall that the Poisson distribution was developed as the limit of a binomial distribution as the number of trials increased to infinity. Consequently, the normal distribution can also be used to approximate probabilities of a Poisson random variable. The approximation is good for λ > 5. Normal Probability Plots How do we know whether a normal distribution is a reasonable model for data? Probability plotting is a graphical method for visual examination of the data. Probability plotting typically uses special graph paper, known as probability paper, that has been designed for the hypothesized distribution. The sample are first ranked for smallest to largest x1, x2 ,…, xn The ordered observations x(j) are then plotted against their observed cumulative frequency (j - 0.5)/n …… on the appropriate probability plot. Normal Probability Plots Example 3-18 Ten observations on the effective service life in minutes of batteries used in a portable personal computer are as follows: 176, 191, 214, 220, 205, 192, 201, 190, 183, 185. We hypothesize that battery life is adequately modeled by a normal distribution. To use probability plotting to investigate this hypothesis Normal Probability Plots A normal probability plot can also be constructed on ordinary graph paper by plotting the standardized normal scores zj against x(j) (j – 0.5)/n = P(Z ≤ zj) = ɸ(zj) Random Samples, Statistics and the Central Limit Theorem 58 Random Samples, Statistics and the Central Limit Theorem Example 3-46 The average tensile strength of eight rubber O-rings was 7274 kPa. Two obvious questions are the following: What can we conclude about the average tensile strength of future O-rings? How wrong might we be if we concluded that the average tensile strength of this future population of O-rings is 7274? Random Samples, Statistics and the Central Limit Theorem x1 x2 ... xn X n E( X ) V (X ) 2 n 60 Random Samples, Statistics and the Central Limit Theorem Example 3-47 Soft-drink cans are filled by an automated filling machine. The mean fill volume is 358 ml, and the standard deviation is 1.5 ml. Assume that the fill volumes of the cans are independent, normal random variables. What is the probability that the average volume of 10 cans selected from this process is less than 355 ml? Random Samples, Statistics and the Central Limit Theorem 62 Random Samples, Statistics and the Central Limit Theorem 63 Questions?