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A practice exam covering topics in Differential Equations, Leslie Matrices, System of Differential Equations, Functions of Two Variables, Probability, and Statistics. The exam includes questions on solving differential equations, finding equilibrium points, computing gradients and eigenvalues, and calculating probabilities.
Typology: Lecture notes
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1.1. Dierential Equation.
1.1.1. #1. Solve the following dierential equation when x =
and y = 1.
dy dx
= − 4 y^4 x
1.1.2. #2. Find the equilibrium points for the dierential equation.
dy dx
= (y + 1)(y − 1)(y − 2)
1.1.3. #3. Which equilibrium points are stable and which are unstable?
1.2. Leslie Matrices. 1
1.2.1. #4. We have a population of newborns, N 0 , and one-year olds, N 1. There are no two-year olds or older. One third of the newborns survive to the next year to be one-year olds. Each newborn produces 1 newborn for the next year.. Each one-year old produces 6 newborns for the next year. What is the Leslie matrix that takes (^) [ N 0 (t) N 1 (t)
to
N 0 (t + 1) N 1 (t + 1)
1.2.2. #5. Find the eigenvalues.
1.2.3. #6. Find the associated eigenvectors.
1.2.4. #7. What is a stable age distribution
(a b
for this population?
2.1. System of Dierential Equations. #8,#9 Solve the system of dierential equations:
x′(t) = 2x(t) + y(t) y′(t) = 7x(t) − 4 y(t) Find the eigenvalues, Find the eigenvectors.
#10 Suppose the previous question had the initial conditions [ x(0) y(0)
Solve the initial value problem. (That is, solve for the c 1 and c 2 )
#11 What type of equilibrium point is at (0, 0)?
#17 Give a parametric equation for the tangent line to the graph of f (in 3-space) for (x, y) = (1, 2) in the direction (− 3 , 4).
#18 Compute the Hessian of f.
#19 Find the one critical point for f.
f (x, y) = 4x^2 + 2x + y^2
#20, #21, #22, #23 Consider the function on the domain x^2 + y^2 ≤ 1. What is the minimum value f takes and where? What is the maximum f takes and where?
3.1. Sandwich. #24 Suppose you have three types of bread (white, wheat, and pita), two types of protein (mushroom and turkey), two types of extras (pickle and tomato), and three types of cheese (provolone, American, and cheddar). Using one type of bread on either end and concerned about the order of ingredients, how many sandwiches can you make?
3.2. Counting. #25 Suppose there are nine females and six males at a dance. How many ways are there to form four couples on the dance oor?
3.3. Standard Deck of Cards. #26a What's the probability of getting a full house when picking ve cards? A full house is a three of a kind and a pair.
Try your hand at other Poker hands. #26b Royal Flush (10 J Q K A of the same suit).
#26c Straight Flush ( ve consecutive cards of the same suit). This includes the straight starting with an ace-low, that is A2345. For simplicity, we'll include the Royal Flush.
#26d Four of a kind
#26e Flush ( ve cards of the same suit). For simplicity, we'll include Royal Flushes and Straight Flushes.
#30 Suppose you have eight green balls, four red balls, and nine blues balls. Pick three balls without replacement. What is the probability you get three balls from two dierent color groups?
#31 Suppose you have ve green balls, two red balls, and three blues balls. Pick two balls with replacement. What is the probability they are the same?
#32 Suppose you have nine green balls, seven red balls, and six blues balls. Pick nine balls without replacement. What is the probability of getting three of each color?
3.7. Some Facts. #33 Give the formula for the probability mass function of a Poisson distribution X with parameter λ = 3? In that case, what E(X) and var(X)?
#34 What is the density function of the standard normal distribution?
#35 State the Poisson Approximation to the Binomial Distribution. Why do we use the approximation? When in practice should we use the Poisson Approximation to approximate the Binomial Distribution? (Just do your best and use your own words.)
#36 Using Chebyshev's Inequality, nd the number of times you'd have to toss a coin to determine the probability of ipping a heads within 0. 05 of its true value with probability at least 0. 95.
#37 State the Central Limit Theorem. (Just do your best and use your own words.)
3.8. Monty Hall. I present some variations and work out the solutions. There may be better solutions, but these will have to do. #38 Classic. There are three doors and one prize. You select a door and then Monty reveals a door which doesn't have a prize. What is the probability of getting a prize if you switch? if you don't?
in your computation of (b). (c) Use the Poisson distribution to approximate the same probability. Use the fact that
e−^5 ≈ 0. 006738
in your computation of (c).
#46, #47, #48 (Version 2) Suppose you are given a coin with probability (^18) of showing up heads. After some time, you have ipped the coin 448 times and have recorded the number of times it came up heads. (a) What is the expected number of times it should come up heads? (b) Using the normal approximation (no histogram adjustment), what is the probability that it comes up heads at least 49 times? If applicable, use the empircal rule. (c) Using the Poisson distribution, what is the probability that it comes up heads at least 49 times? Use the fact that
∑^48
k=
56 k k!
and e−^56 ≈ 4. 78 × 10 −^25
