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Topic 4 - Continuous distributions • Basics of continuous distributions - pages 119 - 124 • Uniform distribution - pages 135 – 136 • Normal distribution - pages 125 - 131 • Gamma distribution - pages 138 - 141 Continuous Random Variables • A continuous random variable can take on values from an entire interval of the real line. • The probability density function (pdf) of a continuous random variable, X, is a function f(x) such that for a < b b P (a X b ) f (x )dx a • The cdf of X is defined as x F (x ) P ( X x ) f (t )dt Some relationships • What is the relationship between f and F? • P(a ≤ X ≤ b) = F(b) – F(a) • P(X = a) = P(a ≤ X ≤ a) = F(a) – F(a) = 0 Pipeline example • A pipeline is 100 miles long and every location along the pipeline is equally likely to break • Let X be the distance measured in miles from the pipeline origin where a break occurs • What is the cdf for X? • What is the pdf for X? • What is P(30 ≤ X ≤ 50)? Requirements of a pdf • A pdf must satisfy the following two requirements: f (x ) 0 for all x f (x )dx 1 • Does the pipeline pdf satisfy these requirements? Uniform distribution • A uniform distribution on the interval from A to B, U(A,B), is defined by a pdf of the form 1 f (x ) B A for A x B • Does f(x) meet requirements? • What is the cdf for the Uniform distribution? Mean and variance of a cont. random variable E (h (X )) h (x ) f (x )dx , expected value of h (X ) X E (X ), mean of X or expected value of X E [(X X ) ], variance of X 2 X 2 E (X ) X 2 X 2 2 M X (t ) E (e ), moment generating function for X tX M X (0) X , M X (0) E (X 2 ) Back to the Uniform • What is the mean of a U(0,1) distribution? • What is the variance of U(0,1) distribution? Gamma distribution • The gamma distribution, G(a,b), is defined by the following pdf 1 f (x ) a x a 1e x / b , x 0,a > 0, b > 0 b G(a ) where G(a ) x a 1e x dx for a > 0. 0 • Properties of the gamma function, G(a) – For a > 1, G(a) (a1)G(a1) – If a is a positive integer, G(a) (a1)! – G(1/2) Properties of the gamma distribution • Is it a valid pdf? • Show M X (t ) 1 (1 b t )a • Show = ab More on the gamma distribution • a is called the shape parameter • b is called the scale parameter • The exponential distribution is a special case of the gamma with a 1. • The gamma distribution is used as a probability model for the time or space before the ath event in a Poisson process where events occur at the rate b1/l. • Gamma calculator Back to the clunker car • Recall that my car breaks down once a week on average. If the breakdowns occur as events in a Poisson process, then what is the probability less than a week passes before my first breakdown? Gamma or Poisson? • Gamma Calculator Pipe example • Defects along a piece of pipe occur as events in a Poisson process with an average of 2 defects every 10 feet. What is the probability that the third defect will occur at least 20 feet from the beginning of the pipe? • Gamma Calculator Normal distribution • The normal distribution, N(,2), has a pdf given by 1 f (x ) e 2 ( x )2 2 2 - x • The normal distribution is always bell shaped. • The normal distribution is defined in terms of its mean and variance (standard deviation). • Normal calculator Weight gain example • The weight gain associated with an antidepressant is normally distributed with a mean of 6 lbs and a standard deviation of 3 lbs. • What is the probability of weight gain? • What is the probability of gaining between 0 and 12 lbs? • Normal Calculator Standard normal distribution • If X has a N(,2) distribution, then Z=(X-)/ has a standard normal distribution, N(0,1). • The standard normal is an important reference distribution. • P(X ≤ x) = P(Z ≤ (x-)/) = F((x-)/) • The cdf of a standard normal, F(z), is tabled in many textbooks • Standardized values, (x-)/, indicate how far in standard deviations the value x is from • For any normal distribution, probabilities can be phrased in terms of standardized values Empirical rule • What is the probability – a normal falls within one standard deviation of the mean? – a normal falls within two standard deviations of the mean? – a normal falls within three standard deviations of the mean? • Normal Calculator Back to the weight gain example • Recall =6 and =3. • Using the empirical rule, answer the following questions: – What is the probability of weight loss? – What is the probability of a weight gain between 0 and 12 pounds? Normal approximations • Normal approximation to Binomial • Normal approximation to Poisson Do my data look normal? • In StatCrunch, a quantile-quantile plot (QQ plot) plots ordered data values versus quantiles of a standard normal distribution. • If the data are from a normal distribution, the points should lie approximately on a straight line. • Concentration data Other distributions • The Weibull distribution and the log normal distribution are used to model failure times. • The beta distribution is used to model proportions. • There are many other distributions out there. • Choose the one that serves as the best probability model for your setting.