Continuous Probability Models - Buisness Management - Lecture Slides, Slides of Business Administration

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Chapter 9

Continuous Probability Models

Outline

• Probability Density Functions (pdfs)
• The Uniform Distribution
• The Exponential Distribution

Probability Density Functions (pdfs)

The key features of pdfs are

1. pdfs never take negative values
2. the area under a pdf is one: P (−∞ < X < ∞) = 1
3. areas under the curve correspond to probabilities
4. P (X ≤ x) = P (X < x) since P (X = x) = 0.

The Uniform Distribution

• Outcomes measured on a continuous scale.
• All outcomes equally likely.

6

Pdf

0 540 x

The Uniform Distribution

General case:

Random variable X with a uniform distribution on a to b has pdf

f (x) =

  

b − a for^ a^ ≤^ x^ ≤^ b 0 otherwise and probabilities can be calculated using the formula

P (X ≤ x) =

    

0 for x < a x − a b − a for^ a^ ≤^ x^ ≤^ b 1 for x > b.

The Uniform Distribution

Mean:

E(X) = μ = (a^ + 2 b)

Variance:

V ar(X) = σ^2 = (b^ −^ a)

2 12

The Exponential Distribution

• used to model lifetimes of products and times between “random” events
• arrivals of orders, customers in a queueing system, ...
• has one (positive) parameter λ

0 2 4 6 8 10

x

Density

The Exponential Distribution

Main features

1. it refers to positive quantities: x > 0
2. larger values of x are increasingly unlikely – exponential decay
3. the value of λ fixes the rate of decay – larger values of λ correspond to more rapid decay.

Probability that the gap is more than 10 minutes is

P (X > 10) = 1 − P (X ≤ 10) = 1 − ( 1 − e−^0.^3 ×^10 )

= e−^0.^3 ×^10 = 0. 050

Probability that the gap is between 5 and 10 minutes is

P (5 < X < 10) = P (X < 10) − P (X ≤ 5) = 0. 950 − 0. 777 = 0. 173

The Exponential Distribution

Mean:

E(X) = μ =^1 λ

Variance:

V ar(X) = σ^2 = (^) λ^12