Probability and Statistics Lecture 8 Dr Ing Erwin

Probability and Statistics Lecture 8 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com 2 0 1 3 President University Erwin Sitompul PBST 8/1

Chapter 6 Some Continuous Probability Distributions President University Erwin Sitompul PBST 8/2

Chapter 6. 1 Continuous Uniform Distribution n |Uniform Distribution| The density function of the continuous uniform random variable X on the interval [A, B] is n The mean and variance of the uniform distribution are and The uniform density function for a random variable on the interval [1, 3] President University Erwin Sitompul PBST 8/3

Chapter 6. 1 Continuous Uniform Distribution Suppose that a large conference room for a certain company can be reserved for no more than 4 hours. However, the use of the conference room is such that both long and short conference occur quite often. In fact, it can be assumed that length X of a conference has a uniform distribution on the interval [0, 4]. (a) What is the probability density function? (b) What is the probability that any given conference lasts at least 3 hours? (a) (b) President University Erwin Sitompul PBST 8/4

Chapter 6. 2 Normal Distribution n Normal distribution is the most important continuous probability distribution in the entire field of statistics. n Its graph, called the normal curve, is the bell-shaped curve which describes approximately many phenomena that occur in nature, industry, and research. n The normal distribution is often referred to as the Gaussian distribution, in honor of Karl Friedrich Gauss, who also derived its equation from a study of errors in repeated measurements of the same quantity. The normal curve President University Erwin Sitompul PBST 8/5

Chapter 6. 2 Normal Distribution n A continuous random variable X having the bell-shaped distribution as shown on the figure is called a normal random variable. n The density function of the normal random variable X, with mean μ and variance σ2, is where π = 3. 14159. . . and e = 2. 71828. . . President University Erwin Sitompul PBST 8/6

Chapter 6. 2 Normal Distribution Normal Curve μ 1 < μ 2, σ 1 = σ 2 μ 1 = μ 2, σ 1 < σ 2 μ 1 < μ 2, σ 1 < σ 2 President University Erwin Sitompul PBST 8/7

Chapter 6. 2 Normal Distribution Normal Curve f(x) v. The mode, the point where the curve is at maximum v. Concave downward v. Point of inflection σ σ v. Concave upward v. Approaches zero asymptotically x μ v. Total area under the curve and above the horizontal axis is equal to 1 President University v. Symmetry about a vertical axis through the mean μ Erwin Sitompul PBST 8/8

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve n The area under the curve bounded by two ordinates x = x 1 and x = x 2 equals the probability that the random variable X assumes a value between x = x 1 and x = x 2. President University Erwin Sitompul PBST 8/9

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve n As seen previously, the normal curve is dependent on the mean μ and the standard deviation σ of the distribution under investigation. n The same interval of a random variable can deliver different probability if μ or σ are different. Same interval, but different probabilities for two different normal curves President University Erwin Sitompul PBST 8/10

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve n The difficulty encountered in solving integrals of normal density functions necessitates the tabulation of normal curve area for quick reference. n Fortunately, we are able to transform all the observations of any normal random variable X to a new set of observation of a normal random variable Z with mean 0 and variance 1. President University Erwin Sitompul PBST 8/11

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve n The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution. President University Erwin Sitompul PBST 8/12

Chapter 6. 3 Areas Under the Normal Curve Table A. 3 Normal Probability Table President University Erwin Sitompul PBST 8/13

Chapter 6. 3 Areas Under the Normal Curve Interpolation n Interpolation is a method of constructing new data points within the range of a discrete set of known data points. n Examine the following graph. Two data points are known, which are (a, f(a)) and (b, f(b)). n If a value of c is given, with a < c < b, then the value of f(c) can be estimated. n If a value of f(c) is given, with f(a) < f(c) < f(b), then the value of c can be estimated. President University Erwin Sitompul PBST 8/14

Chapter 6. 3 Areas Under the Normal Curve Interpolation P(Z < 1. 172)? P(Z < z) = 0. 8700, z = ? President University Erwin Sitompul Answer: 0. 8794 1. 126 PBST 8/15

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given a standard normal distribution, find the area under the curve that lies (a) to the right of z = 1. 84 and (b) between z = – 1. 97 and z = 0. 86. (a) (b) President University Erwin Sitompul PBST 8/16

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given a standard normal distribution, find the value of k such that (a) P ( Z > k ) = 0. 3015, and (b) P ( k < Z < – 0. 18 ) = 0. 4197. (a) (b) President University Erwin Sitompul PBST 8/17

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given a random variable X having a normal distribution with μ = 50 and σ = 10, find the probability that X assumes a value between 45 and 62. President University Erwin Sitompul PBST 8/18

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given that X has a normal distribution with μ = 300 and σ = 50, find the probability that X assumes a value greater than 362. President University Erwin Sitompul PBST 8/19

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given a normal distribution with μ = 40 and σ = 6, find the value of x that has (a) 45% of the area to the left, and (b) 14% of the area to the right. (a) President University Erwin Sitompul PBST 8/20

Chapter 6. 3 Areas Under the Normal Curve Area Under the Normal Curve Given a normal distribution with μ = 40 and σ = 6, find the value of x that has (a) 45% of the area to the left, and (b) 14% of the area to the right. (b) President University Erwin Sitompul PBST 8/21

Chapter 6. 4 Applications of the Normal Distribution A certain type of storage battery lasts, on average, 3. 0 years, with a standard deviation of 0. 5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last less than 2. 3 years. President University Erwin Sitompul PBST 8/22

Chapter 6. 4 Applications of the Normal Distribution In an industrial process the diameter of a ball bearing is an important component part. The buyer sets specifications on the diameter to be 3. 0 ± 0. 01 cm. All parts falling outside these specifications will be rejected. It is known that in the process the diameter of a ball bearing has a normal distribution with mean 3. 0 and standard deviation 0. 005. On the average, how many manufactured ball bearings will be scrapped? President University Erwin Sitompul PBST 8/23

Chapter 6. 4 Applications of the Normal Distribution A certain machine makes electrical resistors having a mean resistance of 40 Ω and a standard deviation of 2 Ω. It is assumed that the resistance follows a normal distribution. What percentage of resistors will have a resistance exceeding 43 Ω if: (a) the resistance can be measured to any degree of accuracy. (b) the resistance can be measured to the nearest ohm only. (a) (b) As many as 6. 68%– 4. 01% = 2. 67% of the resistors will be accepted although the value is greater than 43 Ω due to measurement limitation President University Erwin Sitompul PBST 8/24

Chapter 6. 4 Applications of the Normal Distribution The average grade for an exam is 74, and the standard deviation is 7. If 12% of the class are given A’s, and the grade are curved to follow a normal distribution, what is the lowest possible A and the highest possible B? Lowest possible A is 83 Highest possible B is 82 President University Erwin Sitompul PBST 8/25

Probability and Statistics Homework 7 A 1. Suppose the current measurements in a strip of wire assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 milliamperes 2. (a) What is the probability that a measurement will exceed 13 milliamperes? (b) Determine the value for which the probability that a current measurement is below this value is 98%. (Mo. E 4. 13 -14 p. 113) 2. A lawyer commutes daily from his suburban home to midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3. 8 minutes. Assume the distribution of trip times to be normally distributed. (a) If the office opens at 9: 00 A. M. and the lawyer leaves his house at 8: 45 A. M. daily, what percentage of the time is he late for work? (b) Find the probability that 2 of the next 3 trips will take at least 1/2 hour. (Wa. 6. 15 s. 186) President University Erwin Sitompul PBST 8/26