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Continuous Probability Spaces
• Ω is not countable.
• Outcomes can be any real number or part of an interval of R, e.g. heights,
weights and lifetimes.
• Can not assign probabilities to each outcome and add them for events.
• Define Ω as an interval that is a subset of R.
• F – the event space elements are formed by taking a (countable) number of
intersections, unions and complements of sub-intervals of Ω.
• Example: Ω = [0,1] and F = {A = [0,1/2), B = [1/2, 1], Φ, Ω}
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How to define P ?
• Idea - P should be weighted by the length of the intervals.
- must have P(Ω) = 1
- assign 0 probability to intervals not of interest.
• For Ω the real line, define P by a (cumulative) distribution function as
follows: F(x) = P((- ∞, x]).
• Distribution functions (cdf) are usually discussed in terms of random
variables.
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Recalls
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Cdf for Continuous Probability Space
• For continuous probability space, the probability of any unique outcome
is 0. Because,
P({ω}) = P((ω, ω]) = F(ω) - F(ω) = 0.
• The intervals (a, b), [a, b), (a, b], [a, b] all have the same probability in
continuous probability space.
• Generally speaking,
– discrete random variable have cdfs that are step functions.
– continuous random variables have continuous cdfs.
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Examples
(a) X is a random variable with a uniform[0,1] distribution.
The probability of any sub-interval of [0,1] is proportional to the interval’s
length. The cdf of X is given by:
(b) Uniform[a, b] distribution, b > a. The cdf of X is given by:
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Formal Definition of continuous random variable
• A random variable X is continuous if its distribution function may be
written in the form
for some non-negative function f.
• fX(x) is the (Probability) Density Function of X.
• Examples are in the next few slides….
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The Uniform distribution
(a) X has a uniform[0,1] distribution. The pdf of X is given by:
(b) Uniform[a, b] distribution, b > a. The pdf of X is given by:
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Facts and Properties of Pdf
• If X is a continuous random variable with a well-behaved cdf F then
• Properties of Probability Density Function (pdf)
Any function satisfying these two properties is a probability density
function (pdf) for some random variable X.
• Note: fX (x) does not give a probability.
• For continuous random variable X with density f
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The Exponential Distribution
• A random variable X that counts the waiting time for rare phenomena
has Exponential(λ) distribution. The parameter of the distribution
λ = average number of occurrences per unit of time (space etc.).
The pdf of X is given by:
• Questions: Is this a valid pdf? What is the cdf of X?
• Note: The textbook uses different parameterization λ = 1/β.
• Memoryless property of exponential random variable:
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The Gamma distribution
• A random variable X is said to have a gamma distribution with
parameters α > 0 and λ > 0 if and only if the density function of X is
 e  x x  1

f X x     

0

0 x
otherwise
where
• Note: the quantity г(α) is known as the gamma function. It has the
following properties:
– г(1) = 1
– г(α + 1) = α г(α)
– г(n) = (n – 1)! if n is an integer.
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The Beta Distribution
• A random variable X is said to have a beta distribution with parameters
α > 0 and β > 0 if and only if the density function of X is
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The Normal Distribution
• A random variable X is said to have a normal distribution if and only if,
for σ > 0 and -∞ < μ < ∞, the density function of X is
• The normal distribution is a symmetric distribution and has two
parameters μ and σ.
• A very famous normal distribution is the Standard Normal distribution
with parameters μ = 0 and σ = 1.
• Probabilities under the standard normal density curve can be done using
Table 4 in Appendix 3 of the text book.
• Example:
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Example
• Kerosene tank holds 200 gallons; The model for X the weekly demand is
given by the following density function
• Check if this is a valid pdf.
•
Find the cdf of X.
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Summary of Discrete vs. Continuous Probability Spaces
• All probability spaces have 3 ingredients: (Ω, F, P)
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Poisson Processes
• Model for times of occurrences (“arrivals”) of rare phenomena where
λ – average number of arrivals per time period.
X – number of arrivals in a time period.
• In t time periods, average number of arrivals is λt.
• How long do I have to wait until the first arrival?
Let Y = waiting time for the first arrival (a continuous r.v.) then we have
Therefore,
which is the exponential cdf.
• The waiting time for the first occurrence of an event when the number of
events follows a Poisson distribution is exponentialy distributed.
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Expectation
• In the long run, rolling a die repeatedly what average result do you expact?
• In 6,000,000 rolls expect about 1,000,000 1’s, 1,000,000 2’s etc.
Average is
• For a random variable X, the Expectation (or expected value or mean) of X
is the expected average value of X in the long run.
• Symbols: μ, μX, E(X) and EX.
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Expectation of discrete random variable
• For a discrete random variable X with pmf
whenever the sum converge absolutely
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Examples
1) Roll a die. Let X = outcome on 1 roll. Then E(X) = 3.5.
2) Bernoulli trials
and
. Then
3) X ~ Binomial(n, p). Then
4) X ~ Geometric(p). Then
5) X ~ Poisson(λ). Then
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Expectation of continuous random variable
• For a continuous random variable X with density
whenever this integral converge absolutely.
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Examples
1) X ~ Uniform(a, b). Then
2) X ~ Exponential(λ). Then
3) X is a random variable with density
(i) Check if this is a valid density.
(ii) Find E(X)
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4) X ~ Gamma(α, λ). Then
5) X ~ Beta(α, β). Then
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