Chapter 08 - JMiko- Uniform Distributions

Chapter 8 : 

8.1 Chapter 8 Continuous Probability Distributions

Probability Density Functions… : 

8.2 Probability Density Functions… Unlike a discrete random variable which we studied in Chapter 7, a continuous random variable is one that can assume an uncountable number of values.  We cannot list the possible values because there is an infinite number of them.  Because there is an infinite number of values, the probability of each individual value is virtually 0.

Point Probabilities are Zero : 

8.3 Point Probabilities are Zero Because there is an infinite number of values, the probability of each individual value is virtually 0. Thus, we can determine the probability of a range of values only. E.g. with a discrete random variable like tossing a die, it is meaningful to talk about P(X=5), say. In a continuous setting (e.g. with time as a random variable), the probability the random variable of interest, say task length, takes exactly 5 minutes is infinitesimally small, hence P(X=5) = 0. It is meaningful to talk about P(X ≤ 5).

Probability Density Function… : 

8.4 Probability Density Function… A function f(x) is called a probability density function (over the range a ≤ x ≤ b if it meets the following requirements: f(x) ≥ 0 for all x between a and b, and The total area under the curve between a and b is 1.0 f(x) x b a area=1

Uniform Distribution… : 

8.5 Uniform Distribution… Consider the uniform probability distribution (sometimes called the rectangular probability distribution). It is described by the function: f(x) x b a area = width x height = (b – a) x = 1

Example 8.1(a)… : 

8.6 Example 8.1(a)… The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. Find the probability that daily sales will fall between 2,500 and 3,000 gallons. Algebraically: what is P(2,500 ≤ X ≤ 3,000) ? f(x) x 5,000 2,000

Example 8.1(a)… : 

8.7 Example 8.1(a)… P(2,500 ≤ X ≤ 3,000) = (3,000 – 2,500) x = .1667 “there is about a 17% chance that between 2,500 and 3,000 gallons of gas will be sold on a given day” f(x) x 5,000 2,000

Example 8.1(b)… : 

8.8 Example 8.1(b)… The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. What is the probability that the service station will sell at least 4,000 gallons? Algebraically: what is P(X ≥ 4,000) ? f(x) x 5,000 2,000

Example 8.1(b)… : 

8.9 Example 8.1(b)… P(X ≥ 4,000) = (5,000 – 4,000) x = .3333 “There is a one-in-three chance the gas station will sell more than 4,000 gallons on any given day” f(x) x 5,000 2,000

Example 8.1(c)… : 

8.10 Example 8.1(c)… The amount of gasoline sold daily at a service station is uniformly distributed with a minimum of 2,000 gallons and a maximum of 5,000 gallons. What is the probability that the station will sell exactly 2,500 gallons? Algebraically: what is P(X = 2,500) ? f(x) x 5,000 2,000

Example 8.1(c)… : 

8.11 Example 8.1(c)… P(X = 2,500) = (2,500 – 2,500) x = 0 “The probability that the gas station will sell exactly 2,500 gallons is zero” f(x) x 5,000 2,000

HomeworkOn page 258-2598.4, 8.6, 8.7, 8.8, 8.9, 8.13 (this one is tricky) : 

8.12 HomeworkOn page 258-2598.4, 8.6, 8.7, 8.8, 8.9, 8.13 (this one is tricky)