Expected value

• How to Find the Expected Value of an Exponential Function
  Find E[e^X] by integrating against X's distribution. For beta and gamma random variables, standard formulas give the result and its conditions.
• What Are the Expected Value and Standard Deviation of Dice Rolls?
  For a fair six-sided die, the expected roll is 3.5 and the standard deviation is about 1.708; for n independent rolls, use the scaling rules.
• Expected Value of 1/(X + 1) for a Poisson Distribution
  For X ~ Poisson(λ), E[1/(X+1)] = (1−e^−λ)/λ. See the series derivation, a worked example, and the special case λ = 0.
• Expected Value of a Geometric Distribution by First-Step Analysis
  Use first-step analysis to derive the mean of a geometric waiting time, then see how the answer changes when you count failures instead of trials.
• What Is the Expected Value of a Poisson Distribution?
  For a Poisson random variable with parameter λ, the expected value is λ. See the derivation, a worked rate example, and common mistakes.
• How to Prove the Law of Total Expectation
  The law of total expectation says E[Y] = E[E[Y|X]]. Learn its proof from conditional expectation's definition, with discrete, continuous, and worked numerical examples.
• How to Find an Unbiased Estimator
  An estimator is unbiased when its expected value equals the target for every parameter value; identify the model, compute its expectation, and verify it.
• How to Find the Mean and Variance of an Exponential Distribution
  For an exponential distribution with rate λ, the mean is 1/λ and the variance is 1/λ²; see the derivation and scale-parameter version.
• Why Can’t I Compute the Expected Value from a Uniform MGF?
  For X uniform on [a,b], the MGF has an apparent 0/0 at t=0; extend it by continuity, then differentiate to get E[X]=(a+b)/2.
• Law of Iterated Expectations: A Worked Example
  The law of iterated expectations says to average a conditional mean over the variable you conditioned on; a weighted two-bag example shows each step.
• Mean and Variance of the Weibull Distribution
  For a Weibull distribution with shape k and scale λ, the mean is λΓ(1+1/k) and the variance is λ²[Γ(1+2/k)−Γ(1+1/k)²].
• Expected Value of 1/(1 + X) for a Geometric Random Variable
  For a geometric random variable X, E[1/(1+X)] is -p ln(p)/(1-p) on support starting at zero; starting at one gives a different formula.
• What Is an Unbiased Estimator of a Bernoulli Distribution?
  For independent Bernoulli observations, the sample proportion of successes is an unbiased estimator of the unknown success probability.
• What Is Wald’s Second Equation?
  Wald’s second equation gives the mean squared centered sum at a stopping time: the increment variance times the expected stopping time.