Midterm 1 Extra Questions

Questions from Recent Material

Extra Questions from Chapter 8: 1, 2, 3, 4

Additional Questions

What would be the result from running the following code?
```
all( c(1,2,3,4,5) > 0)
```

Consider the following function

a_function <- function(n) {
  out <- 0
  for (i in 1:n) {
    out <- out + i^2
  }
  out
}

If you run the following code, what will it output?

a_function(5)

Suppose there are two random variables $X$ and $Y$ .
1. If you know that $X$ and $Y$ are independent, do you know what their covariance is equal to? Explain. If yes, what is the covariance equal to?
2. If you know that $cov (X, Y) = 0$ , are $X$ and $Y$ independent? Explain.
3. If you know that $cov (X, Y) = 1$ , are $X$ and $Y$ independent? Explain.
Suppose that $X_{1}$ and $X_{2}$ are two random variables such that $E [X_{1}] = 0$ , $E [X_{2}] = 5$ , $var (X_{1}) = 1$ , $var (X_{2}) = 10$ and $cov (X_{1}, X_{2}) = - 1$ . Suppose that $Y = X_{1} + X_{2}$ .
1. What is $E [Y]$ ?
2. What is $var (Y)$ ?
Consider a random variable $Y$ that is equal to a firm’s profits (in thousands of dollars) and another random variable $X$ that is equal to firm’s number of employees. Suppose you know that $\begin{array}{r} E [Y | X = x] = 50 + 10 x \end{array}$
1. Explain how to interpret $E [Y | X = x]$ .
2. What is $E [Y | X = 10]$ ?
3. Suppose that $var (Y) = 40$ , $E [X] = 30$ , and $var (Y) = 20$ , calculate $E [Y]$ .
Suppose that we have a random sample of $n$ observations of $X$ and $Y$ .
1. Suppose that you want to estimate the covariance between $X$ and $Y$ using the data that we have. Propose an estimator for the covariance. Hint: Try using the analogy principle and the expression $cov (X, Y) = E [X Y] - E [X] E [Y]$ .
2. Alternatively, the definition of covariance is $cov (X, Y) = E [(X - E [X]) (Y - E [Y])]$ . Propose an estimator for the covariance based on this expression. Would you expect this to give you the same estimate of the covariance as in part a?