STATISTICS and PROBABILITY

Transcription

1 Introduction to Statistics Atatürk University STATISTICS and PROBABILITY LECTURE: PROBABILITY DISTRIBUTIONS Prof. Dr. İrfan KAYMAZ Atatürk University Engineering Faculty Department of Mechanical Engineering P.S. These lecture notes are mainly based on the reference given in the last page.

2 objectives of this lecture Introduction tostatistics Atatürk University After carefully listening of this lecture, you should be able to do the following: Determine means and variances for discrete random variables. Determine means and variances for continuous random variables. Binomial distribution Uniform Distribution Normal Distribution Standart Normal Distribution

3 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Mean Defined or E X The mean or expected value of the discrete random variable X, denoted as E X x f x x, is The mean is the weighted average of the possible values of X, the weights being the probabilities where the beam balances. It represents the center of the distribution. It is also called the arithmetic mean. If f(x) is the probability mass function representing the loading on a long, thin beam, then E(X) is the fulcrum or point of balance for the beam. The mean value may, or may not, be a given value of x.

4 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Variance Defined 2 The variance of X, denoted as or V X, is V X E X x f x x f x x The variance is the measure of dispersion or scatter in the possible values for X. It is the average of the squared deviations from the distribution mean. x Figure : The mean is the balance point. Distributions (a) & (b) have equal mean, but (a) has a larger variance.

5 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Mean & Variance Suppose X x probability density function f. The mean or expected value The variance is a continuous random variable with E X xf x dx of X, denoted as or E X, is 2 of X, denoted as V X or, is 2 (4-4) V X x f x dx x f x dx The standard deviat of i 2 ion X s.

6 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Example: Electric Current For the copper wire current measurement, the PDF is f(x) = 0.05 for 0 x 20. Find the mean and variance x E X x f xdx x 10 V X x 10 f xdx

7 Binomial Distribution Definition The random variable X that equals the number of trials that result in a success is a binomial random variable with parameters 0 < p < 1 and n = 0, 1,... The probability mass function is: If X is a binomial random variable with parameters p and n, μ = E(X) = np and σ 2 = V(X) = np(1-p). John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

8 Example What is the probability of getting 4 heads if the experiments are repeated 10 times? What is the probability of getting 6 for 12 times if a die is thrown 20 times? John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

9 Continuous Uniform Distribution This is the simplest continuous distribution and analogous to its discrete counterpart. A continuous random variable X with probability density function f(x) = 1 / (b-a) for a x b Figure : Continuous uniform PDF John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

10 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Mean & Variance Mean & variance are: 2 a b b a E X and V X (4-7)

11 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Example: Uniform Current Let the continuous random variable X denote the current measured in a thin copper wire in ma. Recall that the PDF is F(x) = 0.05 for 0 x 20. What is the probability that the current measurement is between 5 & 10 ma? 10 P 5 x dx

12 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Continuous Uniform CDF F x x a 1 x a du b a b a The CDF is completely described as 0 x a F x x a b a a x b 1 b x Figure : Graph of the Cumulative Uniform CDF

13 Normal Distribution The most widely used distribution is the normal distribution, also known as the Gaussian distribution. Random variation of many physical measurements are normally distributed. The location and spread of the normal are independently determined by mean (μ) and standard deviation (σ). Figure : Normal probability density functions John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

14 Normal Probability Density Function John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

15 Empirical Rule P(μ σ < X < μ + σ) = P(μ 2σ < X < μ + 2σ) = P(μ 3σ < X < μ + 3σ) = Figure : Probabilities associated with a normal distribution John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University 15

16 Standard Normal Distribution A normal random variable with μ = 0 and σ 2 = 1 is called a standard normal random variable and is denoted as Z. The cumulative distribution function of a standard normal random variable is denoted as: Φ(z) = P(Z z) = F(z) John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University 16

17 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Standardizing 2 Suppose X is a normal random variable with mean and variance. X x Then, P X x P P Z z (4-11) where Z is a standard normal random variabl e, and x z is the z-value obtainedby standardizing X. The probability is obtained by using Appendix Table III with z x.

18 Example Assume Z is a standard normal random variable. a) Find P(Z 1.50) b) Find P(Z 1.53) c) Find P(Z 0.02) John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

19 Example: Standard Normal Exercises John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University 1. P(Z > 1.26) = P(Z < -0.86) = P(Z > -1.37) = P(-1.25 < 0.37) = P(Z -4.6) 0 6. Find z for P(Z z) = 0.05, z = Find z for (-z < Z < z) = 0.99, z = 2.58 Figure : Graphical displays for standard normal distributions.

20 Example: Normally Distributed Current-1 With μ = 10 and σ = 2 ma, what is the probability that the current measurement is between 9 and 11 ma? John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

21 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Example: Normally Distributed Current-1 Answer: 9 10 x P9 X 11 P P 0.5 z P z 0.5 P z Figure: Standardizing a normal random variable.

22 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Example: Shaft Diameter-1 The diameter of the shaft is normally distributed with μ = inch and σ = inch. The specifications on the shaft are ± inch. What proportion of shafts conform to the specifications? Let X denote the shaft diameter in inches.

23 Example: Shaft Diameter-1 Answer: P X P Z P 4.6 Z P Z 4.6 P Z Sec 4-6 Normal Distribution 23 John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University

24 Next Week John Wiley & Sons, Inc. Applied Statistics and Probability for Engineers, by Montgomery and Runger. Atatürk University Joint probability functions.

25 Atatürk University References Douglas C. Montgomery, George C. Runger Applied Statistics and Probability for Engineers, John Wiley & Sons, Inc.