Continuous Random Variables and Probability Density Functions

A family of pdf's that has been used to approximate the distribution of income, city population size, and size of firms is the Pareto family. The family has two parameters ... (see attachment)

Continuous Random Variables and Probability Distributions

The Rockwell hardness of a metal is determined by impressing a hardened point into the surface of the metal and then measuring the depth of penetration of the point. Suppose the Rockwell hardness of a particular alloy is normally distributed with mean 70 and standard deviation ... (see attachment)

Normal approximation to a binomial distribution

Suppose that 10% of all steel shafts produced by a certain process are nonconforming but can be re-worked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let X denote the number among these that are nonconforming, what is the (approximate) probability that X is: a. At most 30 b. Less than 3 ...continues

Gamma Distribution

Suppose that when a transistor of a certain type is subjected to an accelerated life test, the lifetime X (in weeks) has a gamma distribution with mean ... (see attachment)

Imaginary ( Complex ) Numbers, Discriminants and Real World Applications

1. When solving a quadratic equation using the quadratic formula, it is possible for the b2 - 4ac term inside the square root (the discriminant) to be negative, thus forcing us to take the square root of a negative number. The solutions to the equation will then be complex numbers (i.e., involve the imaginary unit i). Question ...continues

Jointly Distributed Random Variables

10. Annie and Alvie have agreed to meet between 5:00 pm and 6:00 pm for dinner at a local health food restaurant. Let X=Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly distributed on the interval [5, 6]. a. What is the joint pdf of X and Y? b. What is the probability that ...continues

Jointly Distributed Random Variables : Marginal Probability Density Function

12. Two components of a minicomputer have the following joint pdf for their useful life times X and Y: (see attachment) a. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? Explain. c. What is the probability that the l ...continues

Joint Distribution Table : Compute the expected revenue.

26. Consider a small ferry that can accommodate cars and buses. The toll for cars is $3, and the toll for buses is $10. Let X and Y denote the number of cars and buses, respectively, carried on a single trip. Suppose the joint distribution of X and Y is as given in the table in Exercise 7 (see attachment). Compute the expected r ...continues

Compute the Covariance and Coefficient of Correlation

30. a. Compute the covariance for X and Y in Exercise 22 (see attachment) b. Compute ρ (correlation coefficient) for X and Y in the same exercise. Please see the attached file for the fully formatted problems.

Probabilities related to a sample mean.

The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi. a. What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200. b. If the sample size had been 15 rather than 40, could the probability requested in part (a) b ...continues