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MIT Math Methods for QF > Week 1: Probability > Flashcards

Week 1: Probability Flashcards

1
Q

What is a random variable?

A

A function which assigns a number to events in the sample space.

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2
Q

What is the probability of a random variable taking a given value expressed as?

A

P(X = value)

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3
Q

In the case of continuous random variables, what is the probability of any given value?

A

Zero

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4
Q

What is a probability distribution?

A

A function that satisfies p(x) ≥ 0 and sums to one for discrete variables or integrates to one for continuous variables.

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5
Q

What does the cumulative distribution function (CDF) represent?

A

The probability that X is less than or equal to a given value.

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6
Q

How is the CDF related to the probability density function (PDF)?

A

p(x) = dF(x)/dx

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7
Q

What is the expectation of a random variable?

A

A weighted average where the weight of each event is equal to its probability.

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8
Q

What is the formula for the expected value in the discrete case?

A

E[f(X)] = Σ f(xk)p(xk)

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9
Q

What is the mean of a distribution?

A

The expectation of the random variable itself, µ = E[X]

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10
Q

What are moments of a distribution?

A

The expectation of powers of the random variable.

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11
Q

What does variance measure?

A

Dispersion based on the second moment.

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12
Q

How is variance calculated?

A

Var(X) = E[(X - µ)²]

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13
Q

What are skewness and kurtosis?

A

Skewness measures asymmetry; kurtosis measures tail weights.

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14
Q

What is the covariance of two random variables?

A

Cov(X, Y) = E[(X - µx)(Y - µy)] = E[XY] - µxµy

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15
Q

What does correlation measure?

A

The strength and direction of a linear relationship between two random variables.

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16
Q

What is the range of correlation coefficients?

A

-1 ≤ ρ(X, Y) ≤ 1

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17
Q

What characterizes a uniform distribution?

A

p(x) = 1 for x in [0, 1] and 0 otherwise.

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18
Q

What is the mean of a uniform distribution?

A

µ = 0.5

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19
Q

What defines a binomial distribution?

A

Two possible outcomes: success or failure, with parameters n (number of trials) and p (probability of success).

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20
Q

What is the probability mass function of a binomial distribution?

A

f(k; n, p) = (n choose k) * p^k * q^(n-k) where q = 1 - p.

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21
Q

What is the mean of a binomial distribution?

A

µ = np

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22
Q

What is a Gaussian distribution also known as?

A

Normal distribution

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23
Q

What parameters define a Gaussian distribution?

A

Mean (µ) and variance (σ²)

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24
Q

What is the probability density function of a Gaussian distribution?

A

p(x) = (1/(√(2πσ²))) * e^(-(x-µ)²/(2σ²))

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25
What is a lognormal distribution?
A distribution of a variable whose logarithm is normally distributed.
26
What is the Poisson distribution used for?
To model the number of events occurring in a fixed interval of time or space.
27
What is the probability mass function of a Poisson distribution?
p(k; λ) = (e^(-λ) * λ^k) / k!
28
What is the mean of a Poisson distribution?
µ = λ
29
What is the relationship between binomial and Poisson distributions?
Poisson is the limiting case of the binomial distribution as n approaches infinity and np = λ is fixed.
30
What does 'k!' represent?
Factorial of k ## Footnote k! = k × (k - 1) × (k - 2) × ... × 1
31
What is the relationship between k! and (k - 1)!?
k! = k × (k - 1)! ## Footnote This shows that the factorial of k can be expressed in terms of the factorial of k-1.
32
What does the term 'fat tails' refer to in statistics?
Distributions with large kurtosis or where higher moments diverge ## Footnote This implies that the Central Limit Theorem may not hold.
33
Give an example of a distribution with fat tails.
Cauchy distribution ## Footnote Cauchy distribution is known for having undefined variance and higher moments.
34
What are common probability distributions used in finance?
* Uniform * Binomial * Poisson * Normal (Gaussian) * Lognormal ## Footnote These distributions have well-defined moments and applications in various financial models.
35
What is the sum of random variables S defined as?
S = X1 + X2 + ... + Xn ## Footnote This represents the total of several random variables.
36
How can the expected value of the sum of random variables be calculated?
E[S] = E[X1] + E[X2] + ... + E[Xn] ## Footnote This uses the linearity of expectation.
37
What is the Central Limit Theorem (CLT)?
The sum of a large number of independent random variables will be approximately normally distributed ## Footnote This holds true if individual distributions are well-behaved.
38
What does the variance of a portfolio depend on?
Covariance of pairs of asset returns ## Footnote The portfolio risk is measured by its standard deviation.
39
How is the portfolio return Rp calculated?
Rp = Σ wiRi ## Footnote where wi is the portfolio weight in asset i and Ri is the return on asset i.
40
What is the variance of the portfolio return?
Var(Rp) = Σ wi^2Var(Ri) + 2ΣwiwjCov(Ri, Rj) ## Footnote This includes both the variances of individual assets and their covariances.
41
What is a characteristic function in probability?
Defines the moments of the distribution via its derivatives ## Footnote The characteristic function is useful for sums of random variables.
42
What happens to the sum of N independent Gaussian random variables?
It is also a Gaussian random variable ## Footnote The mean and variance of the resulting Gaussian are derived from the individual means and variances.
43
What is the Law of Large Numbers (LLN)?
As n increases, the probability that the mean deviates from np goes to zero ## Footnote This indicates that sample averages converge to the expected value.
44
What is the scaling variable z in the context of binomial distribution?
z = (k - np) / sqrt(npq) ## Footnote This is used for standardizing the binomial random variable.
45
What does the binomial distribution describe?
The number of successes in a fixed number of Bernoulli trials ## Footnote Each trial has two outcomes: success or failure.
46
How is the variance of the binomial distribution computed?
Var(S) = np(1 - p) ## Footnote This formula indicates the spread of the distribution based on the probability of success.
47
What is the significance of the convolution of probability distributions?
It allows the calculation of the probability distribution of the sum of random variables ## Footnote Convolution is a mathematical operation that combines two functions.
48
What is the primary use of moment generating functions?
To derive the moments of a distribution ## Footnote They can simplify the calculation of expected values and variances.
49
What is the definition of cumulant expansion?
Cumulant expansion is defined as: p˜(t) ⌘ exp(Σ_{n=0}^{∞} (it)^n / n! C_n), where C_n = (-i)^n (d^n / dt^n log p˜(t))|_{t=0} ## Footnote This relates the logarithm of the Fourier transform of a distribution to its cumulants.
50
What are the first four cumulants for a random variable X?
* C1 = hXi * C2 = hX²i - hXi² * C3 = hX³i - 3hXihX²i + 2hXi³ * C4 = hX⁴i - 3hX²i² - 4hXihX³i + 12hXi²hX²i - 6hXi⁴ ## Footnote These cumulants provide information about the shape and characteristics of the distribution.
51
For a Gaussian distribution, what is the value of cumulants for n > 2?
All cumulants are exactly zero for n > 2 ## Footnote This reflects the fact that Gaussian distributions have certain symmetry and characteristics.
52
What does the central limit theorem state about cumulants of independent, identically-distributed random variables?
Cumulants add: Cˆ_n = N * C_n ## Footnote This indicates that the cumulants of the sum of these variables scale with the number of variables.
53
As N approaches infinity, what happens to the cumulants for n > 2?
All cumulants vanish for n > 2 ## Footnote This shows that the distribution approaches a Gaussian shape.
54
What is the behavior of normalized cumulants with respect to N?
Cˆ_n = 1/N^(n/2-1) * C_n ## Footnote This shows a power-law dependence on N.
55
What is the significance of the characteristic function of a probability distribution?
It provides a compact formula for generating all of the moments ## Footnote This is useful for understanding the properties of the distribution.
56
How is the characteristic function related to sums of random variables?
It has simple properties when applied to sums ## Footnote This is particularly relevant in the context of the Central Limit Theorem.
57
What is a unique property of Gaussian distributions regarding their Fourier transforms?
The Fourier transform of a Gaussian is also a Gaussian ## Footnote This property is fundamental in probability theory.
58
What does the Central Limit Theorem (CLT) indicate about the sum of IID random variables?
The sum approaches a Gaussian distribution ## Footnote This demonstrates the universality of the Gaussian distribution under certain conditions.
59
True or False: A sum of Gaussian random variables is also a Gaussian random variable.
True ## Footnote This reinforces the property of Gaussian distributions in probability theory.
60
Fill in the blank: The distribution of the sum of six uniform random variables approaches a _______ as the number of variables increases.
Gaussian ## Footnote This is a direct application of the Central Limit Theorem.

MIT Math Methods for QF (2 decks)

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