Expected value formula: E[X] = Σ [x * P(x)]
Expected value
Expected value formula: E[X] = Σ [x * P(x)]
The expected value formula for a discrete random variable X is the sum of the products of each outcome x and its probability P(x). This formula captures the weighted average of all possible outcomes, where the weights are the probabilities.
Example
For a fair six-sided die, the expected value E[X] is calculated as follows: E[X] = (1*1/6) + (2*1/6) + (3*1/6) + (4*1/6) + (5*1/6) + (6*1/6) = 3.5
Understanding the expected value formula is fundamental in probability theory as it helps in predicting the average outcome of a random variable over many trials.
Related concepts
Poisson distribution
Poisson distribution formula: P(k; λ) = (λ^k * e^(-λ)) / k!
BLEU
BLEU = exp(Σ(w_t * log(p_t)))
Entropy (information theory)
H(X) = −∑x∈X p(x) log(p(x))
Normal distribution
Normal distribution PDF formula
Conditional probability
P(A|B) = P(A ∩ B) / P(B)
Mean squared error
Mean squared error (MSE) formula: MSE = (1/n) * Σ(y_i - ŷ_i)²
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