Transforms associated with uniform random variables. (a) Find the transform associated with an integer-valued random variable X that is uniformly distributed in the range {a, a + 1, . . . , b}. (b) Find the transform associated with a continuous random variable X that is uniformly distributed in the range [a, bl . Solution. (a) The PMF of X is if k = a, a + 1. ... , b, otherwise. The transform is A1(s) = L esk p(X = k) k=-oc (b) We have b L - 1 sk e b-a + l k=a sa b-a _ e e sk b-a + l L..,. k=O e sa e s(b-a+l) - 1 b-a + l sx J b e Sx eS b - e sa M(s) = E[e 1 = a b _ a d x = s(b - a) .

Simple Linear Regression Analyzing relationship between 2 variables X= hours studied Y=test score Response variable (y)= dependent variable Explanatory variable (x)= independent variable o Helper o Helps estimate mean of y’s Scatterplot used to identify o Patterns/ relationships o Outliers What regression line gives you o Line of means for y population o Prediction of the mean o Value of the regression equation is the “least squares” estimate of the mean of that y subpopulation Include units!! sample correlation coefficient, r Measures the strength of the LINEAR association between 2 quantitative variables P= population corr