QTA 2 - RANDOM VARIABLES Flashcards
What is the difference between a probability mass function (PMF) and a cumulative distribution function (CDF)?
PMF assigns probabilities to distinct values of a discrete random variable, while CDF measures the total probability of observing a value less than or equal to a given input.
What are the four common population moments?
- Mean
- Variance
- Skewness
- Kurtosis
What does the quantile function represent?
The quantile function is the inverse of the CDF and defines two moment-like measures: the median and the interquartile range.
How do continuous random variables differ from discrete random variables?
Continuous random variables produce values from an uncountable set, while discrete random variables produce distinct values.
What are the properties of a probability mass function (PMF)?
- Must return non-negative values
- The sum of all probabilities in the support must equal one
What is the formula for the cumulative distribution function (CDF) in relation to PMF?
F_X(x) = Σ f_X(t) for all t in R(X) where t ≤ x.
What is the expected value of a random variable?
The expected value is the weighted average of all possible outcomes, where the weights are the probabilities of those outcomes.
How is the expected value of a Bernoulli random variable calculated?
E[X] = 0 × (1 - p) + 1 × p = p.
What is Jensen’s inequality?
Jensen’s inequality states that for a concave function, E[h(X)] < h(E[X]), and for a convex function, E[g(X)] > g(E[X]).
What is the variance of a random variable?
The variance measures the degree to which the values of a random variable differ from its expected value and is defined as σ² = E[(X - μ)²].
Define skewness in the context of random variables.
Skewness measures the asymmetry of a distribution, calculated as E[(X - μ)³]/σ³.
What does kurtosis indicate about a random variable?
Kurtosis measures the heaviness of the tails of a distribution, with a normal distribution benchmarked at 3.
Fill in the blank: The expected value of a function of a random variable X is defined as E[f(X)] = _______.
Σ f(x) × P(X = x) for x in R(X).
What is the relationship between the CDF and PMF for discrete random variables?
The PMF can be derived from the CDF as f_X(x) = F_X(x) - F_X(x - 1).
What is the expected value of a fair die roll?
E[X] = (1/6) × (1 + 2 + 3 + 4 + 5 + 6) = 3.5.
What is the standard deviation of a random variable?
The standard deviation is the square root of the variance and measures the volatility of a random variable.
True or False: The expectation operator is a nonlinear operator.
False.
What is the support of a discrete random variable?
The set of distinct values that the random variable may take.
What is the expected value of the exponential of a Bernoulli random variable?
E[exp(X)] = (1 - p) + p * exp(1).
What is the expected value of a random variable expressed in a linear combination?
E[cX + a] = cE[X] + a, where c and a are constants.
What does the term ‘support’ refer to in the context of a discrete random variable?
The support refers to the set of values that the random variable may take.
What is the first moment of a random variable?
The first moment is the expected value, denoted as μ₁ = E[X].
Fill in the blank: The expected value of a constant is _______.
the constant itself.
What is the fourth standardized moment known as?
Kurtosis
Kurtosis measures the tails of the distribution.
What is the kurtosis of a normally distributed random variable?
3
Kurtosis greater than 3 indicates heavy-tailed distributions.
What type of distributions are described as heavy-tailed?
Distributions with kurtosis greater than 3
Financial return distributions are often heavy-tailed.
How is a standardized version of a random variable (X) constructed?
Using the formula ( rac{X - mu}{sigma} )
This results in a variable with mean 0 and unit variance.
What do (a) and (b) represent in the linear transformation (Y = a + bX)?
Location shift (a) and scale (b)
(a) affects the mean, and (b) affects the standard deviation.
What is the effect of the location shift (a) on the variance of (Y)?
It has no effect
Variance measures deviations around the mean.
If (b > 0), how do the skewness and kurtosis of (Y) compare to those of (X)?
They are identical
If (b < 0), skewness changes sign, but kurtosis remains unchanged.
What is a continuous random variable?
A variable with a continuous support
It uses a probability density function (PDF) instead of a probability mass function (PMF).
What is the integral property of a probability density function (PDF)?
The integral of the PDF across its support equals 1
This is similar to the summation property of a PMF.
How can the PDF be derived from the cumulative distribution function (CDF)?
By taking the derivative of the CDF
The relationship is ( f_X(x) = rac{dF_X}{dx} ).
What is the expectation (mean) of a continuous random variable (X)?
The integral ( E[X] = int x f_X(x) dx )
This calculates the average value of the variable.
What is the definition of the α-quantile of a random variable (X)?
The smallest number (q) such that (Pr(X < q) = alpha)
This quantile function is denoted by (Q_X(alpha) = F^{-1}(alpha)).
What does the median represent in a data set?
The 50% quantile
It is the middle value when the data is ranked in ascending order.
How is the median calculated when there is an even number of observations?
It is the average of the two middle numbers
This ensures an accurate representation of the central tendency.
What does the interquartile range (IQR) measure?
The difference between the 75% and 25% quantiles
It is a measure of dispersion that is less sensitive to outliers.
Why are quantiles significant for random variables?
They are always well-defined, even for heavy-tailed distributions
Unlike moments, which may not be finite for heavy-tailed variables.
What does the mode measure in a distribution?
The location of the most frequently observed values
In continuous distributions, it is the highest point in the PDF.
