Engineering Statistics ECIV 2305 Section 2.6 Combinations & Functions of Random Variables

Linear Functions of a Random Variable If X is a random variable and Y is a linear function of the random variable X, where Y = aX +b (a & b are numbers) then it is a general rule that: E(Y) = aE(X) +b and Var(Y) = a2 Var(X) (how?)

Linear Functions of a Random Variable An important application of this result will be used in chapter 5, which concerns the “standardization” of a random variable X to have a zero mean and a unit variance. The new “standardized” random variable will be: If you apply the previous linear function rule, then

…Linear Functions of a Random Variable In order to find the pdf of the new transfromed random variable Y, one should do the following: Start with the pdf of the random variable X Find the cdf of the random variable X , i.e. FX(x) Convert from the cdf of X to the cdf of Y i.e. from FX(x) to FY(y) Differentiate the cdf of Y to get the pdf of Y The new part here is how to convert from the cdf of X to the cdf of Y

…Linear Functions of a Random Variable

Example: Test Score Standardization (page 142) Suppose that the raw scores X from a particular testing procedure are distributed between -5 and 20 with an expected value of 10 and a variance of 7. It is required to standardize the scores so that they lie between 0 and 100 using a linear transformation. Find the expected value and the standard deviation of the new standardized scores.

…example: Test Score Standardization (page 142)

Example: Chemical Reaction Temperatures (page 142) The temperature X in degrees Fahrenheit of a particular chemical reaction is known to be distributed between 220o and 280o with a probability density function of With an expectation of E(X) = 255o and a standard deviation of σX = 16.58o. Suppose that the chemist wishes to convert the temperatures to degrees Centigrade. Find the pdf of the temperature in degrees Centigrade along with the new mean and standard deviation in degrees Centigrade

… Example: Chemical Reaction Temperatures (page 142)

… Example: Chemical Reaction Temperatures (page 142)

Linear Combinations of Random Variables If X1, X2, …….. , Xn is a sequence of random variables and a1, a2, ….., an and b are constants, and Y is a linear combination in the following form Y = a1X1 + a2X2 + …….. + anXn + b then E(Y) = a1E(X1) + a2E(X2) + …….. + anE(Xn)+ b and Var(Y) = (a1)2Var(X1)+ (a2)2Var(X2)+…+ (an)2Var(Xn)

Example: page 146 Suppose that X1 and X2 are two independent random variables and both of them have an expectation of μ and a variance of σ2 . and suppose that Y1 and Y2 are two random variable for which Y1 = X1 + X2 Y2 = X1 – X2 Find the expectations and variances of Y1 and Y2. E(Y1)= E(X1)+ E(X2) = 2μ and E(Y2)= E(X1) – E(X2) = zero Var(Y1)= (1)2Var(X1)+ (1)2Var(X2) = 2σ2 and Var(Y2)= (1)2Var(X1)+ (-1)2Var(X2) = 2σ2

Conclusion from the previous example: Adding or subtracting independent random variables increases variability

Averaging Independent Random Variables Suppose that X1 , X2, ……, Xn is a sequence of independent random variables each with an expectation μ and a variance of σ2 , and with an average of