{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

CEE11_s11_ch3

# CEE11_s11_ch3 - Discrete random variables CEE 11 ch 3...

This preview shows pages 1–7. Sign up to view the full content.

Discrete random variables CEE 11 – ch 3 Copyright Dr. Brigitte Baldi 2011 © Random variables A random variable X is a function that associates a unique numerical value with every outcome of a random experiment. A discrete random variable can take only a finite (or a countably infinite, i.e. ordered list) number of values. 0 for head, 1 for tail, when flipping a coin Number of items inspected before a defective item is found Number of cracks in a portion of a highway during the winter months A continuous random variable can take any real value within a finite or infinite interval. Life time, pH, concentration of pollutants, ...

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Probability mass function (PMF) The probability mass function, p(x) = P(X = x) , of a discrete random variable X gives the probability that the random variable will take on each of its possible values. p(x) = 1/16 x = 0 4/16 x = 1 6/16 x = 2 4/16 x = 3 1/16 x = 4 0 otherwise X is the number of heads in 4 successive coin flips
Cumulative distribution function If p(x) = P(X = x) is the probability mass function (pmf) of a discrete random variable, then its cumulative distribution function (cdf) is F(x) = P(X x) = Σ p(y) for y x If a and b are any two numbers such that a b then F(a X b) = F(b) – F(a ־ ) where F(a ־ ) evaluates F at the point immediately to the left of a. p(x) = 1/16 x = 0 4/16 x = 1 6/16 x = 2 4/16 x = 3 1/16 x = 4 0 otherwise X is the number of heads in 4 successive coin flips F(x) = 0 x < 0 1/16 x = 0 5/16 x = 1 11/16 x = 2 15/16 x = 3 16/16 x = 4 1 x > 4 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 X F (x )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Expected value (mean) The expected value E(X) or mean μ of a random variable X represents its central value. It is the weighted sum of the possible values of X in the domain D using the probabilities as weights. p(x) = 1/16 x = 0 4/16 x = 1 6/16 x = 2 4/16 x = 3 1/16 x = 4 0 otherwise X is the number of heads in 4 successive coin flips E(X) = (0)(1/16) + (1)(4/16) + (2)(6/16) + (3)(4/16) + (4)(1/16) = 32/16 = 2 Variance The variance V(X) or σ 2 of a random variable X is a measure of spread about its expected value μ . The standard deviation σ is the square root of σ 2 p(x) = 1/16 x = 0 4/16 x = 1 6/16 x = 2 4/16 x = 3 1/16 x = 4 0 otherwise X is the number of heads in 4 successive coin flips E(X) = 2 V(X) = (0-2) 2 (1/16) + (1-2) 2 (4/16) + (2-2) 2 (6/16) + (3-2) 2 (4/16) + (4-2) 2 (1/16) = (4+4+0+4+4)/16 = 1
Calculations rules for the mean Let X be a random variable taking values over D and with pmf p(x). If Y is a function of X the such that Y = h(X) , then If a and b are two constants, then E(aX) = aE(X) E(X + b) = E(X) + b E(aX + b) = aE(X) + b Also note that, if X and Y are two random variables, then E(X + Y) = E(X) + E(Y) E(X – Y) = E(X) – E(Y) Calculations rules for the variance Let X be a random variable taking values over D and with pmf p(x). If Y is a function of X the such that Y = h(X) , then If a and b are two constants, then V(aX) = a 2 V(X) V(X + b) = V(X) V(aX + b) = a 2 V(X) Also note that, if X and Y are two independent random variables, then V(X + Y) = V(X) + V(Y) V(X – Y) = V(X) + V(Y) { } 2 [ ( )] ( ) [ ( )] ( ) x D V h X h x E h X p x =

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document