Download Homework 9 (optional) 1. A fair coin is tossed 100 times. What does

Probability and Statistics, Spring 2014
Grinshpan
Homework 9
(optional)
1. A fair coin is tossed 100 times. What does the Chebyshev inequality tell you about the probability
that the number of heads that turn up deviates from the expected number by three or more standard
deviations?
2. Let X be a random variable with E[X] = 0 and Var(X) = 1. What integer value k will assure us
that P (|X| ≥ k) ≤ .01?
3. A fair coin is tossed a large number of times n. Does the Law of Large Numbers assure us that, if n
is large enough, with probability > .99 the number of heads that turn up will not deviate from n/2 by
more than 100?
4. Let Sn be the number of heads that turn up in n tosses of a fair coin. Consider the events
En : |Sn /n − .5| < 0.1, n = 1, 2, . . . . What is the smallest value of n such that P (En ) ≥ 0.9?
5. Let Sn be the number of heads that turn up in n tosses of a fair coin. Estimate
P (S100 ≤ 45), P (45 < S100 < 55), and P (S200 = 100).
6. A die is rolled 24 times. Use the Central Limit Theorem to estimate the probabilities that the sum is
greater than 84 and that the sum is equal to 84.
7. A random walker starts at 0 on the x-axis and at each time unit moves 1 step to the right or 1
step to the left with probability 1/2. Estimate the probability that, after 100 steps, the walker is more
than 10 steps from the starting position.
8. Suppose we choose independently 25 numbers at random (uniform density) from the interval [0, 20].
Write the normal densities that approximate the densities of their sum S25 and their average A25 .