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.
