Expected value: the weighted average

Key idea

For a discrete random variable with possible values \(x_1,x_2,\dots\) and probabilities \(p(x_1),p(x_2),\dots\), the expected value is the weighted average: \[ E[X] = \sum_x x\,p(x). \] Think: “multiply each outcome by how often it happens, then add.”

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Summary

• Expected value is a weighted average: \(E[X]=\sum x\,p(x)\).
• It represents the long-run mean, not necessarily a value you can observe in one trial.

Linearity of expectation: your best shortcut

Key idea

Linearity of expectation works always: \[ E[aX+b]=aE[X]+b,\quad E[X+Y]=E[X]+E[Y]. \] You do not need independence for expected values to add.

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Summary

• Linearity: \(E[aX+b]=aE[X]+b\).
• Sums: \(E[X+Y]=E[X]+E[Y]\) (independence not required).

Variance: measure spread around the mean

Key idea

If \(\mu=E[X]\), then: \[ \mathrm{Var}(X) = E[(X-\mu)^2]. \] The most useful shortcut is: \[ \mathrm{Var}(X)=E[X^2]-\mu^2. \] This avoids expanding \((X-\mu)^2\) for every outcome.

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Summary

• Variance measures spread around the mean.
• Fast identity: \(\mathrm{Var}(X)=E[X^2]-(E[X])^2\).

Variance rules: shift, scale, and sums

Key idea

Two essential rules: \[ \mathrm{Var}(aX+b)=a^2\mathrm{Var}(X), \] and for sums: \[ \mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y)+2\mathrm{Cov}(X,Y). \] If \(X\) and \(Y\) are independent, then \(\mathrm{Cov}(X,Y)=0\) and variances add: \[ \mathrm{Var}(X+Y)=\mathrm{Var}(X)+\mathrm{Var}(Y). \]

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Summary

• Shift doesn’t change variance; scaling by \(a\) multiplies variance by \(a^2\).
• For independent sums, add variances.

Common distributions and fastest formulas

Key idea

Some distributions appear everywhere:

• Bernoulli(\(p\)) (one trial, \(X\in\{0,1\}\)): \(E[X]=p\), \(\mathrm{Var}(X)=p(1-p)\).
• Binomial(\(n,p\)) (sum of \(n\) independent Bernoulli trials): \(E[X]=np\), \(\mathrm{Var}(X)=np(1-p)\).
• Uniform(\([0,1]\)): \(E[X]=\tfrac12\), \(\mathrm{Var}(X)=\tfrac{1}{12}\).

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Summary

• Bernoulli: \(E[X]=p\), \(\mathrm{Var}(X)=p(1-p)\).
• Binomial: \(E[X]=np\), \(\mathrm{Var}(X)=np(1-p)\).
• Uniform(\([0,1]\)): \(E[X]=\tfrac12\), \(\mathrm{Var}(X)=\tfrac{1}{12}\).

Fast practice with small distributions

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Summary

• Use \(E[X]\) as a weighted average to find the mean quickly.
• Use \(\mathrm{Var}(X)=E[X^2]-(E[X])^2\) for fast variance.

Why expected value and variance matter

Where these ideas show up

• Games and fairness: expected value tells you a fair entry fee; variance tells you risk.
• Statistics: mean and variance summarize distributions and power confidence intervals.
• Data and simulation: Monte Carlo estimates rely on expected value and measure error with variance.
• Finance and planning: average return vs. volatility (spread) is a mean/variance story.

Worked example: a simple profit game

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Final recap

• Expected value is a weighted average: \(E[X]=\sum x\,p(x)\).
• Linearity: \(E[X+Y]=E[X]+E[Y]\) and \(E[aX+b]=aE[X]+b\).
• Variance: \(\mathrm{Var}(X)=E[(X-\mu)^2]=E[X^2]-\mu^2\).
• Variance rules: \(\mathrm{Var}(aX+b)=a^2\mathrm{Var}(X)\); independent sums add variances.
• Standard deviation is the square root of variance, measured in the same units as \(X\).