Introduction to the Continuous Distributions

The Uniform Distribution

Equally Likely If Y takes on values in an interval (a, b) such that any of these values is equally likely, then To be a valid density function, it follows that

Uniform Distribution A continuous random variable has a uniform distribution if its probability density function is given by

Uniform Mean, Variance Upon deriving the expected value and variance for a uniformly distributed random variable, we find is the midpoint of the interval and

Example Suppose the round-trip times for deliveries from a store to a particular site are uniformly distributed over the interval 30 to 45 minutes. Find the probability the delivery time exceeds 40 minutes. Find the probability the delivery time exceeds 40 minutes, given it exceeds 35 minutes. Determine the mean and variance for these delivery times.

The Normal Distribution

Bell-shaped Density The normal random variable has the famous bell-shaped distribution. The most commonly used continuous distribution. The normal distribution is used to approximate other distributions (see Central Limit Theorem). For a normal distribution with E(Y) = m and V(Y) = s2.

Standard Normal Curve For the standard normal distribution E(Y) = 0 and V(Y) = 1.

Normal Probabilities For finding probabilities, we compute Or, at least approximate the value numerically using an algorithm like Simpson’s Rule (Calc. 1?)

Well-known Areas For a data distribution which is normally distributed (i.e., shaped like a normal curve), certain properties are often quoted. Properties: 68.3 % of the data lies within + 1 std. dev. of the mean. 95.4 % of the data lies within + 2 std. dev. of the mean.

The probability Keep the picture in mind! For mean 16 and standard deviation 0.8,… P(16 < X < 17.8) = ? area of 0.4878, or 48.78% of the data in this interval, between 16 and 17.8.

Using the table Let z equal the distance (in standard deviations) a value x falls from the center. We may compute z as As in our example of the previous page, the value x = 17.8 was located at a distance of … .04 .05 2.0 2.1 0.4878 2.2

Width of Interval Find the percentage (or probability) for the interval 24 < X < 26.4 P(24 < X < 26.4) = 0.4772

Width of Interval Also, provided in table (see Appendix) For 2.00 standard deviations above the mean… .00 .01 .02 2.0 0.4772 2.1 2.2

Example For a normal distribution with m = 30 and s = 2 find percentage of data which falls in the interval between 30 and 33.4. First, sketch a "bell-curve", centered at 30, and shade the region of interest. About 45.5%

Rest of the Half Exactly one-half (an area of 0.5) lies below the mean and half lies above the mean. Find the percentage of data which falls in the interval greater than 33.4.

To the Left of the Mean Also, due to the symmetry, the area within z standard deviations below the mean is the same as z standard deviations above the mean.

Example Suppose the hours spent studying per week for students in normally distributed with a mean of 18.4 hours, standard deviation of 2.5. What percentage of students study more than 20 hours per week?

The Rest of the Half If we determine the area above the mean and less than 20. The "area in the tail" is the rest of the half. 50% - 23.89% = 26.11%

Continuing... This time determine the percentage of students who study less than 22 hours per week?

Backwards? For a standard normal distribution, find z such that P( Z < z ) = 0.8686 For a normal distribution with m = 5 and s = 1.5, find b such that P( Y < b ) = 0.8686 If a soft drink machine fills 16-ounce cups with an average of 15.5 ounces, what is the standard deviation given that the cup overflows 1.5% of the time?

Exponential Distribution A special case of the Gamma Distribution

Time till arrival? Consider W, the time until the first arrival. Number of customers t T W is a continuous random variable. What can we say about its probability distribution?

Inter-arrival times If the average arrivals per unit time equals l, the probability that zero arrivals have occurred in the interval (0, w) is given by the Poisson distribution F(w) = P(W < w) = 1 – P(W > w) Sometimes written where b = 1/ l is the average inter-arrival time (e.g., “minutes per arrival”).

Exponential Distribution A continuous random variable W whose distribution and density functions are given by and is said to have an exponential distribution with parameter (“average”) b .

Exponential Random Variables Typical exponential random variables may include: Time between arrivals (inter-arrival times) Service time at a server (e.g., CPU, I/O device, or a communication channel) in a queueing network. Time to failure (“lifetime”) of a component.

0.2 arrivals per minute Distributions for W, time till first arrival: ( using integration-by-parts ) As expected, since average time is 1/0.2 = 5 minutes/arrival.

Exponential mean, variance If W is an exponential random variable with parameter b, the expected value and variance for W are given by Also, note that

CO concentrations Air samples in a city have CO concentrations that are exponentially distributed with mean 3.6 ppm. For a randomly selected sample, find the probability the concentration exceeds 9 ppm. If the city manages its traffic such that the mean CO concentration is reduced to 2.5 ppm, then what is the probability a sample exceeds 9 ppm?

Memoryless Note P(W > w) = 1 – P(W < w) = 1 – (1 – e-lw) = e-lw Consider the conditional probability P(W > a + b | W > a ) = P(W > a + b)/P(W > a) We find that The only continuous memoryless random variable.

Gamma Distribution The exponential distribution is a special case of the more general gamma distribution: where the gamma function is For the exponential, choose a = 1 and note G(1) = 1.

Gamma Density Curves Gamma function facts:

Exponential mean, variance If Y has a gamma distribution with parameters a and b, the expected value and variance for Y are given by In the case of a = 1, the values for the exponential distribution result.

Recognize the distribution Find E(Y) and V(Y) by inspection given that

Chi-Square Distribution As another special case of the gamma distribution, consider letting a = v/2 and b = 2, for some positive integer v. This defines the Chi-square distribution. Note the mean and variance are given by