
    f <- function(x){(x/4)*exp(-x^2/8)}


  # define the function ex


  # compute the expected value of X


  # define the function ex2


  # compute the variance of X


  # define the function ex
  ex <- function(x){x*f(x)}

  # compute the expected value of X
  expected_value <- integrate(ex, 0, Inf)$value

  # define the function ex2
  ex2 <- function(x){x^2*f(x)}

  # compute the variance of X
  variance <- integrate(ex2, 0, Inf)$value - expected_value^2


  test_function_definition("ex",
                         function_test = {
                           test_expression_result("ex(1)")
                           test_expression_result("ex(4)")
                           test_expression_result("ex(10)")
                           test_expression_result("ex(100)")
                         })
  test_function("integrate", args = c("f", "lower", "upper"), eval = c(F, T, T)) 
  test_object("expected_value")
  test_function_definition("ex2",
                         function_test = {
                           test_expression_result("ex2(1)")
                           test_expression_result("ex2(4)")
                           test_expression_result("ex2(10)")
                           test_expression_result("ex2(100)")
                         })
  test_function("integrate", args = c("f", "lower", "upper"), eval = c(F, T, T))
  test_object("variance")