The Normal Distribution In Reverse Example Given a

The Normal Distribution “In Reverse” • Example: Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X. 1) If P(Z < k) = 0. 45, k = ______ 2) Z = _______ X = _____

Normal Approximation to the Binomial • If n is large and p is not close to 0 or 1, or • If n is smaller but p is close to 0. 5, then the binomial distribution can be approximated by the normal distribution using the transformation: • Look at example 6. 15, pg. 162

Continuous Probability Distributions • Many continuous probability distributions, including: ü Uniform ü Normal ü Gamma ü Exponential ü Chi-Squared ü Lognormal ü Weibull

Gamma & Exponential Distributions • Recall the Poisson Process ü Number of occurrences in a given interval or region ü “Memoryless” process • Sometimes we’re interested in the time or area until a certain number of events occur. • For example An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. § What is the probability that up to a minute will elapse before 2 calls arrive? § How long before the next call?

Gamma Distribution • The density function of the random variable X with gamma distribution having parameters α (number of occurrences) and β (time or region). x > 0. μ = αβ σ = αβ 2

Exponential Distribution • Special case of the gamma distribution with α = 1. x > 0. ü Describes the time until or time between Poisson events. μ=β σ = β 2

Example An average of 2. 7 service calls per minute are received at a particular maintenance center. The calls correspond to a Poisson process. What is the probability that up to a minute will elapse before 2 calls arrive? β = ____ α = ____ P(X ≤ 1) = _________________ How long before the next call?

Your turn … • Look at problem 2, page 174.

Updated homework … • Today’s assignment (due Wednesday): 12, 13, & 15, pp. 157 -158 Add problems 3 & 8, pg. 164 • Wednesday’s assignment (due Friday): Add problem 8, pg. 174 to the assignment