Honors Mathematics I

A coin is tossed 4 times. Let the random variable X denote the number of tails which occur.

1. List the outcomes of the experiment.
2. Find the value assigned to each outcome of the experiment by X.
3. Find the event comprising the outcomes to which a value of 2 has been assigned by X.
4. Find the probability distribution of X.

Let X be a random variable with the probability distribution shown below:

1. Find P(X = 3).
2. Find P(2 < X < 6)
3. Draw a histogram representing the probability distribution of X.
4. Find the expected value of X.
5. Find the standard deviation of X.

Let X be a random variable with a probability distribution where each of the numbers 1, ..., 10 is equally likely to occur.

1. Find the expected value of X.
2. Find the standard deviation of X.
3. Use Chebychev's inequality to estimate the probability that the outcome is within one standard deviation of the expected value.
4. Calculate the exact probability that the outcome is within one standard deviation of the expected value directly from the probability distribution.
5. What range of values, centred about the expected value, will the random variable be in 80% of the time?

Let X be a random variable with a probability distribution where each of the numbers 1, ..., n is equally likely to occur.

1. Find the expected value of X.
2. Find the standard deviation of X.

Find the distance between the points (1,0,0,0,3,4) and (-1,0,2,-2,2,2) in 6 dimension space.

Solve the system of simultaneous linear equations

2x + 3y = 5
x - y = 2

Use Gauss-Jordan elimination to solve the system of simultaneous linear equations:

2x₁ - x₂ - x₃ = 0
3x₁ + 2x₂ + x₃ = 7
x₁ + 2x₂ + 2x₃ = 5

A theatre has a seating capacity of 900 and charges $2 for children, $3 for students and $4 for adults. At a certain screening with full attendance, there were half as many adults as children and students combined. The receipts totalled $2800. How many children attended the show?

Use Gauss-Jordan elimination to solve the system of simultaneous linear equations, or show that there is no solution:

3x₁ - 2x₂ + x₃ = 4
x₁ + 3x₂ - 4x₃ = -3
2x₁ - 3x₂ + 5x₃ = 7
x₁ - 8x₂ + 9x₃ = 10

The flow of traffic in a downtown area is shown by the following diagram:

The numbers and letters represent the number of cars per hour using each section of road, while the arros indicate the direction that traffic flows on each section of road. A road section can handle up to 1000 cars per hour without congestion.

1. Set up a system of linear equations describing the traffic flow. (Hint: the number of cars going in to each intersection must equal the number of cars going out.)
2. Solve the system of equations, and give two solutions in which no road section is congested.
3. Suppose that the section of 7th Ave between 3rd and 4th streets is going to be closed for road work (so that x₆ = 0). Find flow pattern with this restriction which avoids congestion.