Week 2: Random Variables Flashcards
Random Variable
A random variable is a function that takes an outcome from a random experiment (from the sample space
Ω) and assigns it a value in a set E, known as the outcome or target space.
Target Space (E) may be different than sample space if I assign it numbers for representation. Example: in deck of cards, I make Ace = 1, 2 =2, etc.
Real - Valued Random Variable
If the values of a random variable come from real numbers (like height, weight, or temperature), it’s called a real-valued random variable. For example, if X represents the temperature, then X is a real-valued random variable because temperature is a real number.
Vector Valued Random Variable
If the random variable takes multiple values at the same time (like in a list or a point in space), it’s called a vector-valued random variable. For example, if X represents both height and weight together, X would take a pair of values like (height, weight), making it vector-valued.
Discrete Random Variable
If the random variable can only take a countable number of distinct values, it’s called a discrete random variable. For example, rolling a die can result in only 6 distinct values: 1, 2, 3, 4, 5, or 6. This makes it a discrete random variable.
Statistical Analysis and Pushforward Measure
Instead of studying the actual random event, we focus on the probability distribution of the random variable—basically, how likely different values of the random variable are. For example, if you roll a die, we are more interested in the chances of getting a certain number (e.g., 1/6 chance of rolling a 3) than the actual physical act of rolling the die.
What is a Preimage of a set?
The preimage of a set under a random variable is the collection of outcomes in the sample space that lead to a particular result. For example, if you want to know the probability of rolling an even number on a die, the preimage is the set of outcomes that give an even number (i.e., rolling 2, 4, or 6).
(X^(−1)) (B)
Means pre-image of the set B under the random variable X. Identifies all outcomes in the random space that correspond to values in B. For example, if you want to know the probability of rolling an even number on a die, the preimage is the set of outcomes that give an even number (i.e., rolling 2, 4, or 6).
B (subset of target space)
Represents a subset of the target space (or range) of the random variable X. This set includes specific values that you are interested in, such as outcomes that meet certain criteria.
For example, if X is a random variable representing the results of rolling a die, B could be the set {3,4}.
∈
The element-of symbol indicates membership in a set. For example, ω∈Ω means that ω is an element of the set Ω. (Omega is a sample space)
Ω
Represents the sample space, which is the set of all possible outcomes of a random experiment. It defines the universe of all outcomes that can occur.
For instance, if you roll a die, Ω={1,2,3,4,5,6}
X(ω)
This denotes the value of the random variable X at the outcome ω. It maps the outcome from the sample space Ω to the target space E where the values of the random variable lie. This is the actual outcome of the random experiment.
(X^−1)(B) := {ω∈Ω∣X(ω)∈B}
The pre-image of the set B under the random variable X is defined as the set of all outcomes ω in the sample space Ω such that the value of the random variable X at ω is an element of the set B.
B ⊂ E
B is a subset of E (subset of target space)
Px(B)=P(X∈B)
Probability of Random Variable X taking on a value within the set B is equal to the notation Px(B)
What is a Probability Mass Function?
The PMF, denoted px (x), is a function that gives the probability that a discrete random variable X takes a specific value x. It maps each value x in the target space E to its probability.
What does px(x) represent?
The PMF (probability mass function) px(x) = P(X=x). It is the probability that the random variable X equals a specific value x.
Properties of the PMF
Non negativity: for each possible outcome x, within the target space E, probabilities cannot be negative (px(x) >= 0)
Normalization: total probability of all possible outcomes must equal 1 (the sum)
Therefore px(x) is between 0 and 1.
Probability Density Function
A PDF is a function that describes the likelihood of a continuous random variable falling within a certain range. f(x) >= 0 for all x and the total area under the curve is 1.
Continuous Random Variable
A random variable X is continuous if there is a PDF, fX(x), such that the probability of X being between two values (a and b) is given by the integral of fX(x) from a to b.
What is the probability that a continuous random variable X takes on an exact value, i.e., P(X = x)
For a continuous random variable, P(X = x) = 0. This is because the probability at a single point is 0, as given by the integral of the PDF from x to x.
