Some Continuous Probability Distributions

By: Prof. Gevelyn B. Itao

Probability and StatisticsContinuous Uniform Distribution The density function of the continuous uniform

random variable X on the interval [A, B] is

Probability and StatisticsTheorem 6.1The mean and variance of the continuous uniform distribution

f (x; k) are

and

Probability and StatisticsContinuous Uniform DistributionExample 5.1: 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 conferences 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?

Probability and StatisticsContinuous Uniform DistributionExample 5.2: Calculate the mean and variance in prob 1.

Probability and StatisticsNormal DistributionProperties of Normal Curve

1. The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x =

Probability and StatisticsNormal DistributionProperties of Normal Curve

2. The curve is symmetric about a vertical axis through the mean

Probability and StatisticsNormal DistributionProperties of Normal Curve

3. The curve has its points of inflection at x = , is concave downward if - < X < + , and is upward otherwise

Probability and StatisticsNormal DistributionProperties of Normal Curve

4. The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean.

Probability and StatisticsNormal DistributionProperties of Normal Curve

5. The total area under the curve and above the horizontal axis is equal to 1.

Probability and StatisticsAreas Under the Normal Curve

Probability and StatisticsAreas Under the Normal Curve

Probability and StatisticsDefinition 6.1Standard Normal Distribution The distribution of a normal random variable with

mean zero and variance 1.

Probability and StatisticsAreas Under the Normal CurveExample 6.3: Given a standard normal distribution, find the area under the curve that liesa. to the right of z = 1.84b. between z = -1.97 and z = 0.86

Probability and StatisticsAreas Under the Normal CurveExample 6.3: Given a standard normal distribution, find the area under the curve that liesa. to the right of z = 1.84

Probability and StatisticsAreas Under the Normal CurveExample 6.3: Given a standard normal distribution, find the area under the curve that liesb. between z = -1.97 and z = 0.86

Probability and StatisticsAreas Under the Normal CurveExample 6.4: Given a standard normal distribution, find the value of k such thata. P (Z > k) = 0.3015b. P (k < Z < -0.18) = 0.4197

Probability and StatisticsAreas Under the Normal CurveExample 6.4: Given a standard normal distribution, find the value of k such thata. P (Z > k) = 0.3015

From table A.3:

Probability and StatisticsAreas Under the Normal CurveExample 6.4: Given a standard normal distribution, find the value of k such thatb. P (k < Z < -0.18) = 0.4197

From table A.3:

Probability and StatisticsAreas Under the Normal CurveExample 6.5: Given a normal distribution with = 50 and = 10, find the probability that X assumes a value between 45 and 62.

Probability and StatisticsAreas Under the Normal CurveExample 6.5: Given a normal distribution with = 50 and = 10, find the probability that X assumes a value between 45 and 62.

Probability and StatisticsAreas Under the Normal CurveExample 6.5: Given a normal distribution with = 50 and = 10, find the probability that X assumes a value between 45 and 62.

Probability and StatisticsAreas Under the Normal CurveExample 6.6: Given a normal distribution with = 40 and = 6, find the value of x that hasa. 45% of the area to the leftb. 14% of the area to the right

Probability and StatisticsAreas Under the Normal CurveExample 6.6: Given a normal distribution with = 40 and = 6, find the value of x that hasa. 45% of the area to the left

From table A.3:

Probability and StatisticsAreas Under the Normal CurveExample 6.6: Given a normal distribution with = 40 and = 6, find the value of x that hasb. 14% of the area to the right

Probability and StatisticsAreas Under the Normal CurveExample 6.7: A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins.Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given mouse will live a. more than 32 months;b. less than 28 months;c. between 37 and 49 months.

Probability and StatisticsAreas Under the Normal CurveExample 6.8: The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.a. What proportion of rings will have inside diameters

exceeding 10.075 centimeters?b. What is the probability that a piston ring will have

an inside diameter between 9.97 and 10.03 centimeters?

c. Below what value of inside diameter will 15% of the piston rings fall?

Probability and StatisticsAreas Under the Normal Curve

Example 6.9: A lawyer commutes daily from his suburban home to his 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. What is the probability that a trip will take at least 1/2 hour?b. If the office opens at 9:00 A.M. and he leaves his house at 8:45

A.M. daily, what percentage of the time is he late for work?c. If he leaves the house at 8:35 A.M. and coffee is served at the

office from 8:50 A.M. until 9:00 A.M., what is the probability that he misses coffee?

d. Find the length of time above which we find the slowest 15% of the trips.

e. Find the probability that 2 of the next 3 trips will take at least 1/2 hour.

Probability and StatisticsNormal Approximation to the BinomialTheorem 6.2If X is a binomial random variable with mean = np and variance 2 = npq, then the limiting form of the distribution of

as n , is the standard normal distribution n (z; 0,1)

Probability and Statistics

Example 6.10: A process for manufacturing an electronic component is 1% defective. A quality control plan is to select 100 items from the process, and if none are defective, the process continues. Use the normal approximation to the binomial to finda. the probability that the process continues for the

sampling plan described;b. the probability that the process continues even if the

process has gone bad (i.e., if the frequency of defective components has shifted to 5.0% defective).

Normal Approximation to the Binomial

Probability and Statistics

Example 6.11: A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the number of defectivesa. exceeds 13?b. is less than 8?

Normal Approximation to the Binomial

Probability and StatisticsGamma 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.

Probability and StatisticsDefinition 6.2The gamma function is defined by

where α > 0.

Probability and StatisticsTheorem 6.3The mean and variance of the gamma distribution are

andµ = αβ σ2 = αβ2

Probability and Statistics

Example 6.12: In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a random variable X having a gamma distribution with mean µ = 6 and variance σ2 = 12.a. Find the values of α and β.b. Find the probability that on any given day the daily

power consumption will exceed 12 million kilowatthours.

Normal Approximation to the Binomial

Probability and Statistics

Example 6.13: Suppose that the time, in hours, taken to repair a heat pump is a random variable X having a gamma distribution with parameters α =2 and β =1/2. What is the probability that the next service call will requirea. at most 1 hour to repair the heat pump?b. at least 2 hours to repair the heat pump?

Normal Approximation to the Binomial

Probability and StatisticsExponential Distribution The continuous random variable X has an

exponential distribution, with parameter β, if its density function is given by

where β > 0.

Probability and StatisticsTheorem 6.4The mean and variance of the exponential distribution are

andµ = β σ2 = β2

Probability and Statistics

Example 6.14: The life, in years, of a certain type of electrical switch has an exponential distribution with an average life β = 2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?

Normal Approximation to the Binomial