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Discrete random variables
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Discrete random variables take finitely many values, with probability distributions given in tables or expressions where the probabilities sum to 1, and their expected value is found by E(X)=∑xP(X=x), with fair games and linear transformations E(aX+b)=aE(X)+b, Var(aX+b)=a2Var(X).
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Concept of a random variable
SL 4.7
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.
Examples
Human height
The sum of a two dice roll
The number of goals scored in a game
Concept of a discrete random variable
SL 4.7
A discrete random variable takes from a finite set of values:
X∈{x1,x2…xn}
where each possible value has an associated probability.
Discrete probabilities sum to 1
SL 4.7
The sum of the probabilities for all possible values {x1,x2,…xn} of a discrete random variable X equals 1. In symbols,
P(U) =P(X=x1)+P(X=x2)+...+P(X=xn)=x∑ P(X=x)=1
where U denotes the sample space.
Discrete probability distributions in a table
SL 4.7
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.
Discrete probability distributions as an expression
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Probability distributions can be given in a table or as an expression. As an expression, distributions have the form
P(X=x)=(expression in x),x∈{set of possible x}
for any discrete random variable X.
Expected Value
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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:
E(X)=∑x⋅P(X=x)📖
The expected value is often denoted μ.
Fair Games
SL 4.7
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.
Linear transformations of a random variable
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If you add a constant b to every possible value a random variable can take on, its expected value will increase by that constant, reflecting a "shift" in the random variable's distribution. Its variance will remain unchanged, since adding a constant has no impact on the "spread." Multiplying by a constant a scales both expectation and variance:
E(aX+b)Var(aX+b)=aE(X)+b=a2Var(X)
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