The mean IQ of a randomly selected student from a specific

According to the Bureau of Engraving and Printing, http://www.moneyfactory.com/document.cfm/18/106, December 10, 2003, the average life of a one-dollar Federal Reserve note is 22 months. Show that at most 37% of one-dollar bills last for at least five years.

Show that if for a nonnegative random variable X, P (X < 2) = 3/5, then E(X) 4/5.

Let X be a nonnegative random variable with E(X) = 5 and E(X2) = 42. Find an upper bound for P (X 11) using (a) Markovs inequality, (b) Chebyshevs inequality

The average and standard deviation of lifetimes of light bulbs manufactured by a certain factory are, respectively, 800 hours and 50 hours. What can be said about the probability that a random light bulb lasts, at most, 700 hours?

Suppose that the average number of accidents at an intersection is two per day. (a) Use Markovs inequality to find a bound for the probability that at least five accidents will occur tomorrow. (b) Using Poisson random variables, calculate the probability that at least five accidents will occur tomorrow. Compare this value with the bound obtained in part (a). (c) Let the variance of the number of accidents be two per day. Use Chebyshevs inequality to find a bound on the probability that tomorrow at least five accidents will occur.

The average IQ score on a certain campus is 110. If the variance of these scores is 15, what can be said about the percentage of students with an IQ above 140?

The waiting period from the time a book is ordered until it is received is a random variable with mean seven days and standard deviation two days. If Helen wants to be 95% sure that she receives a book by certain date, how early should she order the book?

Show that for a nonnegative random variable X with mean , P (X 2) 1/2.

Suppose that X is a random variable with E(X) = Var(X) = . What does Chebyshevs inequality say about P (X > 2)?

From a distribution with mean 42 and variance 60, a random sample of size 25 is taken. Let X be the mean of the sample. Show that the probability is at least 0.85 that X (38, 46).

The mean IQ of a randomly selected student from a specific university is ; its variance is 150. A psychologist wants to estimate . To do so, for some n, she takes a sample of size n of the students at random and independently and measures their IQs. Then she finds the average of these numbers. How large a sample should she choose to make at least 92% sure that the average is accurate within 3 points?

For a distribution, the mean of a random sample is taken as estimation of the expected value of the distribution. How large should the sample size be so that, with a probability of at least 0.98, the error of estimation is less than 2 standard deviations of the distribution?

To determine p, the proportion of time that an airline operator is busy answering customers, a supervisor observes the operator at times selected randomly and independently from other observed times. Let Xi = 1 if the ith time the operator is observed, he is busy; let Xi = 0 otherwise. For large values of n, if (1/n)/n i=1 Xis are taken as estimates of p, for what values of n is the error of estimation at most 0.05 with probability 0.96?

For a coin, p, the probability of heads is unknown. To estimate p, for some n, we flip the coin n times independently. Let pJ be the proportion of heads obtained. Determine the value of n for which pJestimates p within 0.05 with a probability of at least 0.94.

Let X be a random variable with mean . Show that if E 4 (X )2n 5 < , then for > 0, P ! |X | " 1 2n E 4 (X )2n5 .

Let X be a random variable and k be a constant. Prove that P (X > t) E(e kX) e kt .

Prove that if the random variables X and Y satisfy E 4 (X Y )2 5 = 0, then with probability 1, X = Y .

Let X and Y be two randomly selected numbers from the set of positive integers {1, 2, . . . , n}. Prove that (X, Y ) = 1 if and only if X = Y with probability 1. Hint: First prove that E 4 (X Y )2 5 = 0; then use Exercise 17.

Let X be a random variable; show that for > 1 and t > 0, P * X 1 t ln , 1 MX(t).

Let the probability density function of a random variable X be f (x) = xn n! ex , x 0. Show that P (0 < X < 2n + 2) > n n + 1 . Hint: Note that # 0 xnex dx = 0(n + 1) = n!. Use this to calculate E(X) and Var(X). Then apply Chebyshevs inequality.

Let {x1, x2, . . . , xn} be a set of real numbers and define x = 1 n .n i=1 xi, s2 = 1 n 1 .n i=1 (xi x) 2 . Prove that at least a fraction 1 1/k2 of the xis are between x ks and x + ks. Sketch of a Proof: Let N be the number of x1, x2, . . . , xn that fall in A = [ x ks, x + ks]. Then s2 = 1 n 1 .n i=1 (xi x) 2 1 n 1 . xi,A (xi x) 2 1 n 1 . xi,A k2 s2 = n N n 1 k2 s2 . This gives (N 1)/(n 1) 1 (1/k2). The result follows since N/n (N 1)/(n 1).