Chapter 6

2. Chapter 6 Some Continuous Probability Distributions Chapter 6 Some Continuous Probability Distributions

3. Chapter 6.1 Continuous Uniform Distribution Continuous Uniform Distribution • |Uniform Distribution| The density function of the continuous uniform random variable X on the interval [A,B] is • The mean and variance of the uniform distribution are and • The uniform density function for a random variable on the interval [1, 3]

4. Chapter 6.1 Continuous Uniform Distribution 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]. What is the probability density function? What is the probability that any given conference lasts at least 3 hours?

5. Chapter 6.2 Normal Distribution Normal Distribution • Normal distribution is the most important continuous probability distribution in the entire field of statistics. • Its graph, called the normal curve, is the bell-shaped curve which describes approximately many phenomena that occur in nature, industry, and research. • 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

6. Chapter 6.2 Normal Distribution Normal Distribution • Acontinuous random variable X having the bell-shaped distribution as shown on the figure is called a normal random variable. • The density function of the normal random variable X, with mean μ and variance σ2, is where π=3.14159... and e=2.71828...

7. Chapter 6.2 Normal Distribution Normal Curve • μ1 < μ2, σ1 = σ2 • μ1 = μ2, σ1 < σ2 • μ1 < μ2, σ1 < σ2

8. Chapter 6.2 Normal Distribution Normal Curve f(x) • The mode, the point where the curve is at maximum • Concave downward • Point of inflection σ σ • Concave upward • Approaches zero asymptotically x μ • Symmetry about a vertical axis through the mean μ • Total area under the curve and above the horizontal axis is equal to 1

9. Chapter 6.3 Areas Under the Normal Curve Area Under the Normal Curve • The area under the curve bounded by two ordinates x = x1 and x = x2 equals the probability that the random variable X assumes a value between x = x1 and x = x2.

10. Chapter 6.3 Areas Under the Normal Curve Area Under the Normal Curve • As seen previously, the normal curve is dependent on the mean μ and the standard deviation σ of the distribution under investigation. • 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

11. Chapter 6.3 Areas Under the Normal Curve Area Under the Normal Curve • The difficulty encountered in solving integrals of normal density functions necessitates the tabulation of normal curve area for quick reference. • 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 variableZ with mean 0 and variance 1.

12. Chapter 6.3 Areas Under the Normal Curve Area Under the Normal Curve • The distribution of a normal random variable with mean 0 and variance 1 is called a standard normal distribution.

13. Chapter 6.3 Areas Under the Normal Curve Table A.3 Normal Probability Table

14. Chapter 6.3 Areas Under the Normal Curve Interpolation • Interpolation is a method of constructing new data points within the range of a discrete set of known data points. • Examine the following graph. Two data points are known, which are (a,f(a)) and (b,f(b)). • If a value of c is given, with a<c<b, then the value of f(c) can be estimated. • If a value of f(c) is given, with f(a)<f(c)<f(b), then the value of c can be estimated.

15. Chapter 6.3 Areas Under the Normal Curve Interpolation • P(Z<1.172)? Answer: 0.8794 • P(Z<z)=0.8700, z=? 1.126

16. 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.

17. 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.

18. 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.

19. 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.

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.

21. 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.

22. Chapter 6.4 Applications of the Normal Distribution 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.

23. Chapter 6.4 Applications of the Normal Distribution 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?

24. Chapter 6.4 Applications of the Normal Distribution Applications of the Normal Distribution A certain machine makes electrical resistors having a mean resistance of 40Ω and a standard deviation of 2Ω. It is assumed that the resistance follows a normal distribution. What percentage of resistors will have a resistance exceeding 43Ω if: the resistance can be measured to any degree of accuracy. the resistance can be measured to the nearest ohm only. • 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

25. Chapter 6.4 Applications of the Normal Distribution 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

26. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial • The probabilities associated with binomial experiments are readily obtainable from the formula b(x;n, p) of the binomial distribution or from the table when n is small. • For large n, making the distribution table is not practical anymore. • Nevertheless, the binomial distribution can be nicely approximated by the normal distribution under certain circumstances.

27. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial • If X is a binomial random variable with mean μ = np and variance σ2 = npq, then the limiting form of the distribution of as n ∞, is the standard normal distribution n(z;0,1). • Normal approximation of b(x;15,0.4) • Each value of b(x;15,0.4) is approximated by P(x–0.5<X<x+0.5)

28. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial • Normal approximation of and

29. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial • The degree of accuracy, that is how well the normal curve fits the binomial histogram, will increase as n increases. • If the value of n is small and p is not very close to 1/2, normal curve will not fit the histogram well, as shown below. • The approximation using normal curve will be excellent whenn is large or n is small with p reasonably close to 1/2. • As rule of thumb, if bothnp and nq are greater than or equal to 5, the approximation will be good.

30. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial • Let X be a binomial random variable with parameters n and p. For large n, X has approximately a normal distribution with μ = np and σ2 = npq = np(1–p) and and the approximation will be good if np and nq = n(1–p) are greater than or equal to 5.

31. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial The probability that a patient recovers from a rare blood disease is 0.4. If 100 people are known to have contracted this disease, what is the probability that less than 30 survive? • After interpolation • Can you calculate the exact solution?

32. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial A multiple-choice quiz has 200 questions each with 4 possible answers of which only 1 is the correct answer. What is the probability that sheer guess-work yields from 25 to 30 correct answers for 80 of the 200 problems about which the student has no knowledge?

33. Chapter 6.5 Normal Approximation to the Binomial Normal Approximation to the Binomial PU Physics entrance exam consists of 30 multiple-choice questions each with 4 possible answers of which only 1 is the correct answer. What is the probability that a prospective students will obtain scholarship by correctly answering at least 80% of the questions just by guessing? • It is practically impossible to get scholarship just by pure luck in the entrance exam

34. Chapter 6.6 Gamma and Exponential Distributions Gamma and Exponential Distributions • There are still numerous situations that the normal distribution cannot cover. For such situations, different types of density functions are required. • Two such density functions are the gamma and exponential distributions. • Both distributions find applications in queuing theory and reliability problems. • The gamma function is defined by for α>0.

35. Chapter 6.6 Gamma and Exponential Distributions Gamma and Exponential Distributions • |Gamma Distribution| The continuous random variable X has a gamma distribution, with parameters α and β, if its density function is given by where α>0 and β>0. • |Exponential Distribution| The continuous random variable X has an exponential distribution, with parameter β, if its density function is given by where β>0.

36. Chapter 6.6 Gamma and Exponential Distributions Gamma and Exponential Distributions • Gamma distributions for certain values of the parameters α and β • The gamma distribution with α=1 is called the exponential distribution

37. Chapter 6.6 Gamma and Exponential Distributions Gamma and Exponential Distributions • The mean and variance of the gamma distribution are and • The mean and variance of the exponential distribution are and

38. Chapter 6.7 Applications of the Gamma and Exponential Distributions Applications Suppose that a system contains a certain type of component whose time in years to failure is given by T. The random variable T is modeled nicely by the exponential distribution with mean time to failure β = 5. If 5 of these components are installed in different systems, what is the probability that at least 2 are still functioning at the end of 8 years? • The probability whether the component is still functioning at the end of 8 years • The probability whether at least 2 out of 5 such component are still functioning at the end of 8 years

39. Chapter 6.7 Applications of the Gamma and Exponential Distributions Applications Suppose that telephone calls arriving at a particular switchboard follow a Poisson process with an average of 5 calls coming per minute. What is the probability that up to a minute will elapse until 2 calls have come in to the switchboard? • β is the mean time of the event of calling • α is the quantity of the event of calling

40. Chapter 6.7 Applications of the Gamma and Exponential Distributions Applications Based on extensive testing, it is determined that the average of time Y before a washing machine requires a major repair is 4 years. This time is known to be able to be modeled nicely using exponential function. The machine is considered a bargain if it is unlikely to require a major repair before the sixth year. Determine the probability that it can survive without major repair until more than 6 years. What is the probability that a major repair occurs in the first year? • Only 22.3% survives until more than 6 years without major reparation • 22.1% will need major reparation after used for 1 year

41. Chapter 6.8 Chi-Squared Distribution Chi-Squared Distribution • Another very important special case of the gamma distribution is obtained by letting α = v/2 and β = 2, where v is a positive integer. • The result is called the chi-squared distribution, with a single parameter v called the degrees of freedom. • The chi-squared distribution plays a vital role in statistical inference. It has considerable application in both methodology and theory. • Many chapters ahead of us will contain important applications of this distribution.

42. Chapter 6.8 Chi-Squared Distribution Chi-Squared Distribution • |Chi-Squared Distribution|The continuous random variable X has a chi-squared distribution, with v degrees of freedom, if its density function is given by where v is a positive integer. • The mean and variance of the chi-squared distribution are and

43. Chapter 6.9 Lognormal Distribution Lognormal Distribution • The lognormal distribution is used for a wide variety of applications. • The distribution applies in cases where a natural log transformation results in a normal distribution.

44. Chapter 6.9 Lognormal Distribution Lognormal Distribution • |Lognormal Distribution|The continuous random variable X has a lognormal distribution if the random variable Y = ln(X) has a normal distribution with mean μ and standard deviation σ. The resulting density function of X is • The mean and variance of the chi-squared distribution are and

45. Chapter 6.9 Lognormal Distribution Lognormal Distribution Concentration of pollutants produced by chemical plants historically are known to exhibit behavior that resembles a log normal distribution. This is important when one considers issues regarding compliance to government regulations. Suppose it is assumed that the concentration of a certain pollutant, in parts per million, has a lognormal distribution with parameters μ = 3.2 and σ = 1. What is the probability that the concentration exceeds 8 parts per million? • F denotes the cumulative distribution function of the standard normal distribution • a.k.a. the area under the normal curve

46. Probability and Statistics Homework 7 Suppose the current measurements in a strip of wire are assumed to follow a normal distribution with a mean of 10 milliamperes and a variance of 4 milliamperes2. (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.E4.13-14 p.113) 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) (a) Suppose that a sample of 1600 tires of the same type are obtained at random from an ongoing production process in which 8% of all such tires produced are defective. What is the probability that in such sample 150 or fewer tires will be defective?(Sou18. CD6-13) (b) If 10% of men are bald, what is the probability that more than 100 in a random sample of 818 men are bald?