Content
Discrete random variables
Introduction and properties of random variables, probability distributions, expected value and variance, linear transformations of r.v.'s
Introduction and properties of random variables, probability distributions, expected value and variance, linear transformations of r.v.'s
No exercises available for this concept.
A random variable is a variable that can take different values, each associated with some probability, arising from a random process or phenomenon. We usually denote random variables with a capital letter, such as X.
It can be discrete (taken from a finite set of values) or continuous (taking any value within an interval).
The probability distribution of a random variable tells us how likely each outcome is.
A discrete random variable takes from a finite set of values:
where each possible value has an associated probability.
The sum of the probabilities for all possible values {x1,x2,…xn} of a discrete random variable X equals 1. In symbols,
where U denotes the sample space.
Probability distributions of discrete random variables can be given in a table or as an expression. As a table, distributions have the form
where the values in the row P(X=x) sum to 1.
Probability distributions can be given in a table or as an expression. As an expression, distributions have the form
for any discrete random variable X.
The expected value of a discrete random variable X is the average value you would get if you carried out infinitely many repetitions. It is a weighted sum of all the possible values:
The expected value is often denoted μ.
In probability, a game is a scenario where a player has a chance to win rewards based on the outcome of a probabilistic event. The rewards that a player can earn follow a probability distribution, for example X, that governs the likelihood of winning each reward. The expected return is the reward that a player can expect to earn, on average. It is given by E(X), where X is the probability distribution of the rewards.
Games can also have a cost, which is the price a player must pay each time before playing the game. If the cost is equal to the expected return, the game is said to be fair.