Texas A&M University
Department of Mathematics
Volodymyr Nekrashevych
Fall 2011
Math 411 — Problem Set 10
Issued: 11.29
Due: training
10.1. Show that if X is exponential(1), then Y = X/λ is exponential(λ).
10.2. Suppose that X and Y have joint density f (x, y) = c(x + y) for 0 <
x, y < 1. Find c and P (X + Y < 1/2).
10.3. Suppose X and Y have joint density f (x, y) = 6xy 2 when x, y ≥ 0 and
x + y ≤ 1. Are X and Y independent?
10.4. Suppose a point (X, Y ) is chosen at random from the disk x2 + y 2 ≤ 1.
Find the marginal density of X and the conditional density of Y given
X = x.
10.5. Suppose that 10% of a certain brand of jelly beans are red. Use the
normal approximation to estimate the probability that in a bag of 400
jelly beans there are at least 45 red ones.
10.6. A die is rolled repeatedly until the sum of the numbers obtained is
larger than 200. What is the probability that you need more that 66
rolls to do this?
10.7. Among 625 randomly chosen Swedish citizens, it was found that 25
had previously been citizens of another country. Find a 95% confidence
interval for the true proportion.
10.8. Suppose we take a poll of 2,500 people. What percentage should
the leader have for us to be 99% confident that the leader will be the
winner?
10.9. In a 60-day period in Ithaca 12 days were rainy. Is this observation
consistent with the belief that the true proportion of rainy days in 1/3?