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Linear Algebra for MATH 308
Lecture 1: Vectors, Linear Independence, and Spanning Sets
Lecture 2: Operations with Matrices and Vectors
Lecture 3: Systems of Equations
Lecture 4: Determinant
Lecture 5: Eigenvectors and Eigenvalues
Lecture 6: Matrix Inverses and Diagonalization
Lecture 7: Systems of Differential Equations
Lecture 8: Systems of Differential Equations
Math for Quantitative Finance
Several Variables Calculus
Section 1: Functions of Several Variables
Section 2: Limits and Continuity
Section 3: Partial Derivatives
Section 4: Tangent Planes and Linear Approximations
Section 5: The Chain Rule
Section 6: Directional Derivatives and the Gradient Vector
Section 7: Maximum and Minimum Values
Section 8: Lagrange Multipliers
Differential Equations
Section 1: Integrating Factor
Section 2: Separable Equations
Section 3: Compound Interest
Section 4: Variation of Parameters
Section 5: Systems of Ordinary Differential Equations
Section 6: Matrices
Section 7: Systems of Equations, Linear Independence, and Eigenvalues & Eigenvectors
Section 8: Homogeneous Linear Systems with Constant Coefficients
Section 9: Complex Eigenvalues
Section 10: Fundamental Matrices
Section 11: Repeated Eigenvalues
Section 12: Nonhomogeneous Linear Systems
Mathematical Probability
Section 1: Probabilistic Models and Probability Laws
Section 2: Conditional Probability, Bayes’ Rule, and Independence
Section 3: Discrete Random Variable, Probability Mass Function, and Cumulative Distribution Function
Section 4: Expectation, Variance, and Continuous Random Variables
Section 5: Discrete Distributions
Section 6: Continuous Distributions
Section 7: Joint Distribution Function, Marginal Probability Mass Function, and Uniform Distribution
Section 8: Independence of Two Random Variables, Covariance, and Correlation
Section 9: Conditional Distribution and Conditional Expectation
Section 10: Moment Generating Function
Section 11: Markov’s Inequality, Chebyshev’s Inequality, and Weak Law of Large Numbers
Section 12: Convergence and the Central Limit Theorem
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Mathematical Probability
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Section 1: Probabilistic Models and Probability Laws
Section 1: Probabilistic Models and Probability Laws
Instructions
This section covers the concepts listed below.
For each concept, there is a conceptual video explaining it followed by videos working through examples.
When you have finished the material below, you can go to the
next section
or return to the
main Mathematical Probability page
.
Concepts
Kolmogorov’s Axioms
Discrete Uniform Probability Spaces
Complement Rule
Monotonic Property and Inclusion & Exclusion
Links & Resources
Return to Mathematical Probability Page
Return to Mini-Course Main Page
Kolmogorov's Axioms
Watch Concepts Video
Examples
Directions:
You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
Suppose you have a biased coin in which heads is 3 times more likely to occur than tails. If you flip this coin, what is the probability of getting heads? tails?
Reveal Answer
\(P(\{H\})=\dfrac{3}{4} \mbox{ and } P(\{T\})=\dfrac{1}{4}\)
Watch Video Solution
Flip a fair coin until a tail comes up. Establish the probabilistic model for such a random
experiment.
Reveal Answer
See video for further explanation.
Watch Video Solution
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
What is a probability space? Can you state the axioms?
What is the difference between an outcome and an event in a probabilistic model? To which one do we assign probability?
Does
$P(A)=0$
imply
$A=\mathrm{\varnothing}$
? Can you provide a counterexample, if not?
Can you describe
$\bigcup _{n=1}^{\mathrm{\infty}}\text{}[\frac{1}{n},1]$
and
$\bigcap _{n=1}^{\mathrm{\infty}}(-\frac{1}{n},\frac{2}{n})$
?
Does
$\phantom{\rule{1em}{0ex}}}{\displaystyle \cdot \phantom{\rule{-10pt}{0ex}}\bigcup _{n=1}^{\mathrm{\infty}}[\frac{1}{n},1]$
make sense?
Discrete Uniform Probability Spaces
Watch Concepts Video
Examples
Directions:
You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
We roll a pair of fair dice. What is the probability of getting a sum of 10?
Reveal Answer
\(\dfrac{1}{12}\)
Watch Video Solution
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
Can we have an infinite discrete uniform probability space?
Does "rolling a pair of fair dice and recording the product of the two numbers as the outcome" define a uniform probability space?
Complement Rule
Watch Concepts Video
Examples
Directions:
You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
Roll a fair die four times. What is the probability that some number appears more than once?
Reveal Answer
\(\dfrac{13}{18}\)
Watch Video Solution
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
What does the Complement Rule say, and why is it useful?
What is the complement of the following event in the random experience of rolling two dice:
$A=$
the event of getting a sum of at least 4? Determine all the outcomes of
${A}^{c}$
.
Monotonic Property and Inclusion & Exclusion
Watch Concepts Video
Examples
Directions:
You should attempt to solve the problems first, and then you can check your solution and watch the video for an explanation.
15% of the population in a town is blond, 25% has blue eyes, and 2% is blond with blue eyes. What is the probability that a randomly chosen individual is not blond and does not have blue eyes?
Reveal Answer
62%
Watch Video Solution
There are 30 red balls, 20 green, and 10 yellow balls in an urn. Draw 7 balls without replacement. What is the probability that exactly 2 red or exactly 3 yellow balls are in the sample?
Reveal Answer
\(P(A\cup B) = \displaystyle \frac{ {30 \choose 2 } {30 \choose 5}}{{60 \choose 7}}+ \frac{\binom{10}{3} \binom{50}{4}}{\binom{60}{7}} - \frac{\binom{30}{2} \binom{10}{3} \binom{20}{2}}{\binom{60}{7}}\)
Watch Video Solution
Self-Assessment Questions
Directions:
The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
State the monotonic property of probability.
When do we have
$P(A\cup B)=P(A)+P(B)$
?
⇧ Return to Mathematical Probability Page
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