Associations AE-9

Important

This application exercise is due on 21 Oct at 2:00pm.

library(tidyverse) # for data manipulation and plots
library(praise) # for good vibes

Chapter 6: Association between quantitative variables

  1. If the covariance between \(x\) and \(y\) is \(0\), then the correlation between \(x\) and \(y\) is \(0\) as well

    1. True
    2. False

  2. A retailer calculated the correlation line between the price of an item (\(x\)) and the amount sold (\(y\)). The correlation line is the same if the \(x\) and \(y\) variables are exchanged.

    1. True
    2. False
praise()
  1. If the correlation between number of customers and sales in dollars in retail stores is \(r=0.6\), then what would be the correlation if the sales were measured in thousands of dollars? In euros? (1 euro is worth about \(\$US0.97\))

Chapter 10: Association between random variables

  1. Mix and match

    1. Consequence of positive covariance a. \(p(x, y)\)
    1. Covariance between X and Y
    		 b. \(\rho\)
    1. Property of uncorrelated random variables
    		 c. \(p(x, y) = p(x)p(y)\)
    1. Weighted sum of two random variables
    		 d. \(\rho \sigma_X \sigma_Y\)
    1. Sharpe ratio of a random variable
    		 e. \(Var(X+Y) > Var(X) +Var(Y)\)
    1. Implies \(X\) and \(Y\) are independent random variables
    		 f. \(p(x) = p(y)\)
    1. Implies \(X\) and \(Y\) are identically distributed
    		 g. \(X_1, X_2, X_3\)
    1. Symbol for correlation between random variables
    		 h. \(Var(X, Y) = Var(X) +Var(Y)\)
    1. Symbol for a joint probability distribution
    		 i. \(S(Y)\)
    1. Sequence of iid random variables
    		 j. \(3X - 2Y\)

  2. Independent random variables \(X\) and \(Y\) have the means and standard deviations as given in the following table. Use these parameters to find the expected value and SD of the following random variables that are derived from \(X\) and \(Y\)

    1. \(2X - 100\)

    2. \(0.5Y\)

    3. \(X+Y\)

    4. \(X-Y\)

      Mean SD
      \(X\) 1000 200
      \(Y\) 2000 600 
praise()

  1. Repeat the calculations above but now \(X\) and \(Y\) are not independent and have \(Cov(X, Y) = 12,500\).

  2. What’s the covariance between a random variable \(X\) and a constant?

  3. A student budgets $60 weekly for gas and quick meals off-campus. Let \(X\) denote the amount spent for gas and \(Y\) the amount spent for quick meals in a typical week. Assume the student sticks to the budget.

    1. Can we model \(X\) and \(Y\) as independent random variables? Why or why not?

    2. Suppose we assume \(X\) and \(Y\) are dependent. What is the effect of this dependence on the variance of \(X+Y\)?