CIVL 3103

Continuous Distributions

Learning Objectives - Continuous Distributions

  Define continuous distributions, and identify common distributions applicable to engineering problems.

  Identify the appropriate distribution (i.e. uniform, exponential, normal) for use in solving a problem.

  Apply continuous distribution models to solve engineering-oriented problems.

Probability Density Function

x

f(x)

a b

�

P(a ≤ X ≤ b) = f (x)dxa

b

∫

Probability Density Function •  The probability density function has to satisfy two

conditions: –  and

–  .

�

f (x) ≥ 0

�

f (x) =1−∞

∞

∫

Cumulative Distribution Function

• 

�

F(x) = P(X ≤ x) = f (u)du−∞

x

∫

0

1

x

F(x)

F(b)

b

F(a)

a

�

P(X ≤ a) = F(a)

Continuous Distributions •  The probability that the random

variable X will take on a range of values is:

•  Expected Value:

•  Variance: �

P(a ≤ X ≤ b) = F(b) − F(a)

�

E(x) = µx = xf (x)−∞

∞

∫ dx

�

V (x) = σ x2 = (x − µx )

2 f (x)dx = E(X 2) − E(X)[ ]2−∞

∞

∫

Important Continuous Distributions

• Uniform Distribution

•  Exponential Distribution

• Normal Distribution

The Uniform Distribution

�

f (x) = 1b − a

for a ≤ x ≤ b; (0 elsewhere)

The Uniform Distribution •  Cumulative Distribution Function:

•  Expected Value:

•  Variance:

�

F(x) = x − ab − a

for a ≤ x ≤ b

�

E(x) = µ = a + b2

�

V (x) = σ 2 = 112(b − a)2

Example: The Uniform Distribution

Most calculators can generate “random numbers” as can most spreadsheets (e.g., Excel uses the function RAND to put a random number in a cell). These electronically generated “random numbers” appear, statistically, to be random. By this we mean that we cannot conclude, using various statistical tests, that these numbers do not follow a uniform distribution. The numbers are not really random, though, because they are generated by a computer algorithm that uniquely determines the next number to be generated from the last number generated or from some “seed value” entered by the user or stored in memory. For example, a very simple random number generator is:

This generates pseudo-random numbers between 0 and 1, which we abbreviate as X ~ U [ 0 , 1 ] meaning “the random variable X is uniformly distributed over the range [0,1].”

What is the probability that the random number generator (RNG) above will generate a value between 0.2 and 0.3 the next time it is invoked?

�

xi = (xi−1 + π )5 − int xi−1 + π( )5[ ]

The Exponential Distribution

•  Frequently used to model time between successive events (arrivals or failures).

•  Models the continuous “unit” versus the discrete event (Poisson).

x

f(x)

λ

0

�

f (x) = λe−λx

The Poisson Distribution

λ=1 λ=10

The Exponential Distribution

The Exponential Distribution •  Cumulative Distribution Function:

•  Expected Value:

•  Variance:

�

F(x) =1− e−λx

�

E(x) = µ = 1λ

�

V (x) = σ 2 = 1λ2

Example: The Exponential Distribution

•  Telephone calls arrive at the help desk of a small computer software company at the rate of 15 per hour. What is the probability that the next call arrives within 3 minutes?

Example: 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 the length, X, of a conference has a uniform distribution.

a. What is the probability density function? b. What is the probability that any given conference

lasts at least 3 hours?

Example: 2. The net weight in pounds of a packaged

chemical herbicide is uniform for 49.75<x<50.25 pounds.

a.  Determine the mean and variance of the weight of packages.

b. Determine P(X<50.1).

Example: 3.The time between calls to a plumbing

supply business is exponentially distributed with a mean time between calls of 15 minutes.

a.  What is the probability that there are no calls within a 30-minute interval?

b.  What is the probability that at least one call arrives within a 10-minute interval?

Example: 4.The time to failure (in hours) of fans in a

personal computer can be modeled by an exponential distribution with λ = 0.0003.

a.  What proportion of the fans will last at least 10,000 hours?

b.  What proportion of the fans will last at most 7,000 hours?