Ch. 2 Probability

1. Ch. 2 Probability Random Variables and Probability Distributions

2. Random Variables (r.v.) • Definition: A real valued function on a sample space. • If the outcome of interest has an element of uncertainty such that its value can only be stated probabilistically, the outcome is a r.v. • Can be either: • Discrete-the set of values the variable can assume is finite • Continuous-the set of values the variable can assume is infinite. • Any function of a r.v. is also a r.v. i.e. if X is r.v., Z=g(X) is a r.v..

3. Univariate Probability Distributions:Discrete Random Variables • If a r.v. X can only take on values of x1, x2,…xn with probabilities fx(x1), fx(x2)….fx(xn) and S fx(xi) =1, then X is a discrete r.v. • Figure shows the probability distribution function (pdf) for a discrete r.v.

4. Discrete Random Variables:Cumulative Probability Distribution (cdf) • The cdf represents the probability that X is less than or equal to xk. • The probability that X=xi can be determined from:

5. Univariate Probability DistributionsContinuous Random Variables • Can take on any value in a range of values permitted by the physical process involved. • Probability distribution function (pdf) of a continuous r.v. X is a smooth curve and is denoted by px(x). • The cumulative probability distribution (cdf) is denoted by Px(x) and represents the probability that X is less than or equal to x.

6. Continuous Random Variables • The pdf and cdf functions of continous r.v. are related by: or

7. Definition of a pdf for a Continous Random Variable • A function px(x) defined on a real line can be a pdf IF AND ONLY IF 1. 2.

8. Continuous Random Variables • By definition px(x)=0 outside the range of X. • Px(xl)=0 and Px(xu)=1 where xl and xu are the lower and upper limits of X. • For many distributions these limits are -∞ to ∞ (normal distribution) or 0 to ∞ (lognormal distribution).

9. Continuous Random Variables • The probability that X takes on a value between a and b is given by: and is the area under the pdf between a and b. • The probability that a random variable takes on any particular value from a continuous distribution is 0.

10. PDFs and CDFs

11. Example: Definition of a pdf • Evaluate the constant a for the following expression to be considered a probability function. What is the probability that a value selected at random from this distribution will be a) less than 3? b) fall between 2 and 4? c) be larger than 5? d) be larger than 8?

12. Piecewise continuous distributions • Piecewise continuous distributions satisfying the requirements for a pdf in which the prob(X=d) is not zero are possible. • Can be defined by • Where: and P1(x) and P2(x) are nondecreasing functions of x. • The prob(X=d) would be the magnitude of the jump DP at X=d or is equal to P2(d) – P1(d). • Any finite number of discontinuities of this type are possible.

13. Piecewise continuous pdf

14. Relationship between relative frequency and probability • Let px(x) be the pdf of X. The probability that a single trial of the experiment will result in an outcome between X=a and X=b is given by: • In N independent trials of the experiment the expected number of outcomes in the interval a to be would be: • And the expected relative frequency of outcomes in the interval a to b is:

15. In general if xi represents the midpoint of an interval of X given by The expected relative frequency of outcomes in this interval of repeated, independent trials of the experiment is given by: Since the RHS represents the area under px(x) between it can be approximated by

16. Relationship between relative frequency and probability • If N independent observations of X are available, the actual relative frequency of outcomes in an interval of width Dxi centered on xi, may not equal fxi because x is a r.v. that can only be described probabilistically. • The most probable outcome will equal the observed outcome only if px(x) is the actual pdf of X and for an infinitely large number of observations.