Homework 2

1. Suppose a group of 75 individuals has the following compositon:

   • 7 vegans

   • 12 ovo-lacto vegetarians (so the 7 vegans and this group are exclusive from each other)

   • 56 meat eaters

Suppose as well that the following holds:

1. • The probability a vegan has a heart attack is 0.01

   • The probability an ovo-lacto vegetarian has a heart attack is 0.02

   • The probability a meat eater has a heart attack is 0.03.

What is the probability that a randomly selected person from this group has a heart attack?

2. You have 6 cages, each housing a rat on one end and a piece of cheese on the other. You know the probability a rat will eat the cheese within 1 hour is 0.9. If each rat operates independently of each other rat, what is the probability that exactly 4 of the 6 rats will have eaten the cheese within 1 hour?

3. Suppose you develop a way to test for presence of heavy metals in drinking water. You know that 3% of all drinking water contains heavy metals. Your test has the following properties:

   • There are never any false negatives.

   • The false positive rate is 5%

What is the probability that heavy metals are present in the drinking water given two positive tests in a row? Assume that the results are independent of each other

4. A game is played where you have a 12/20 chance to lose 5 dollars, a 4/20 chance to gain 4 dollars, a 3/20 chance to win 8 dollars, and a 1/20 chance to win 16 dollars. Let X be the random amount won/lost. Write the distribution of X as a table.

5. Suppose a company has the following salary grades and distribution:

   • Grade 1: $30,000, 28% of employees are this grade.

   • Grade 2: $42,000, 35% of employees are this grade

   • Grade 3: $65,000, 21% of employees are this grade

   • Grade 4: $81,000, 10% of employees are this grade

   • Grade 5: $104,000, 5% of employees are this grade

   • Grade 6: $240,000, 1% of employees are this grade.

What is the average salary of a random employee?

6. In #5, if there was a 25% pay cut for every salary grade, what would the new average salary be?

7. Let X be a continuous random variable on [0,2] with pdf f(x) = Ce^(-x/2). Find C.

8. Let X be a continuous random variable on [1,4] with pdf f(x) = 1/(2 sqrt(x)). Find the CDF of X, F(x).

9. In #8 use the CDF to compute P(25/16 < X < 49/16)

10. In #8, compute E(X).