Expected value weights possible values by their probabilities.
Key knowledge
For a discrete real-valued random variable X with probabilities p(x), its finite expected value is Σx xp(x) when Σx |x|p(x) is finite. The expected value need not itself be a possible observed value.
Values and equations
E[X] = Σₓ x p(x)
X
Discrete real-valued random variable
x
Value in the support of X
p(x)
Probability P(X=x)
E[X]
Expected value if Σₓ |x|p(x)<∞
Conditions and scope
Sum over the finite or countable support and require absolute convergence for a finite value.
An expected value describes the probability distribution, not a guaranteed result of one trial.
Sources · 2
References
Sources retain their recorded scope. A source link alone does not establish verification of every statement.
Discrete weighted mean and finite-existence condition.
2026-10-10 · Checked positive/negative-part discussion and discrete examples; no unrestricted convergence claim.
Foundational review
2026-10-10 · Fresh primary-source inspection of the complete compact record, including every statement, equation, quantity and material condition, followed by independent AI editorial review. This check does not certify all linked lessons or every passage of every cited publication.
Sources from other places
Agreement within this scope
Discrete expected value as the absolutely convergent probability-weighted sum.
The notes distinguish undefined sums and allow extended positive infinity; the repository states the finite case.
For a discrete real-valued random variable X with probabilities p(x), its finite expected value is Σx xp(x) when Σx |x|p(x) is finite. The expected value need not itself be a possible observed value.
For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)]. If both variances are positive, Corr=Cov/√(Var X Var Y) lies in [−1,1]. Independence gives zero covariance; its converse generally fails.
M_X(t)=E[eᵗˣ] is an MGF when it is finite on an open interval about zero. Its derivatives at zero yield moments; in particular EX=M′(0) and Var X=M″(0)−[M′(0)]².
For X with density f and measurable g, E[g(X)]=∫R g(x)f(x)dx when ∫|g(x)|f(x)dx<∞. For nonnegative g the expectation may instead be infinite. Finite expectations add linearly without assuming independence.
For a collection containing N individuals with N_i in each observed species, the expected species count in a uniformly drawn subset of n individuals without replacement is E[S_n]=Σ_i[1−C(N−N_i,n)/C(N,n)]. This standardizes a within-collection sampling comparison. It does not estimate all unseen species or remove spatial and detectability biases automatically.
Expected richness is a finite expected species count computed from subset probabilities.