Simple trigonometric proof, with a twist.

Prove: sin(2arcsin(x))=2x*sqrt(1-x^2)

Patterns leading to Solutions

The following problem shows how working out examples and looking for patterns can lead to a general solution. A kindly prison warden decided to free his 25 prisoners for good behavior. The prisoners were locked in separate cells, numbered 1 through 25. Each cell had a lock that opened when the key was turned once and locke ...continues

Probability: Continuous Random Variables

1.) Suppose we are producing copper wire and putting the wire on spools. Each spool contains 100 feet of wire. Defects such as nicks in the wire can occur at random locations. What would be a reasonble distribution for each of the following: (a) the number of spools produced until a spool is produced that contains one or more de ...continues

Discrete random variable with probability mass function

1) Let X be a discrete random variable with probability mass function Pr{X=k}= c/(1+(k^2)) for k= -2,-1, 0, 1, 2. a) determine Pr{x <= 0} (b) Determine the mean of X (c) Determine all medians of X (d) Compute Pr{X=2 | X >= 0} (e) Determine the cumulative distribution function

Tangent to a curve: Condition

Find the condition that the curve y = mx + c to be a tangent to the parabola y^2 = 4 ax and also determine the point of contact.

How to get a million by spelling a word whose letters' product=1000000

If we assign number values to letters in the following way: A = 26, B = 25, C = 24 and so on until Y = 2 and Z =1, spell a word such that the product of its letters is as close to a million as possible. Explain how you went about solving this problem.

Getting three thousand bananas across a one thousand mile desert

You have three thousand bananas that you have to get to a destination 1000 miles away. You can only carry 1000 bananas at a time. You also must eat 1 banana per mile for energy. Assuming you design your trip as efficiently as possible, how many bananas will you have left when you arrive at the destination? (apparently someon ...continues

Statistics: Queueing Problem

A fast food outlet has an average of 8 cars at the drivethrough during "lunch rush" 11am-1pm. On average, 2 cars per min. arrive at the resaurant parking lot, and consider the drivethrough but 25% of the time, an arriving car does not actually enter the drive-through line (i.e. it "balks"). Assume no car enters the line without ...continues

Probability: Moment Generating Functions and Poisson Process

1.) Let X be a discrete random variable with probability mass function Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2. a) Determine c. b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X=2 | X >= 0} g) Determine the moment genera ...continues

Random Variables : Continuous R.V., Exponenetial, R.V, Mean and Variance

3) Let X be a continuous random variable with probability density function f(s)= c(1 + s^2) for -2 <= s <= 2. a) Determine c b) Determine Pr {X <= 0} c) Determine the mean of X d) Why is the previous answer fairly obvious? e) Determine the variance of X f) Compute Pr {X = 2 | X = 0} g) Determine the cumulative distribut ...continues