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⬅ Probability distributions
Probability distributions Expectation and variance of a discrete random variable

Probability distributions — Expectation and variance of a discrete random variable 22 problems

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📘 Notes

Expectation and Variance of a Discrete Random Variable

The expectation and variance describe the average value and spread of a probability distribution.

1. Expected Value (Mean)

The expected value \(E(X)\) is the long-run average value of the random variable.

To calculate:

  • Multiply each value \(x\) by its probability
  • Add the results

2. Variance

The variance measures the spread of the probability distribution.

The standard deviation is:

\[ \sigma = \sqrt{\mathrm{Var}(X)} \]

3. Example

\(x\) 0 1 2
\(P(X=x)\) 0.2 0.5 0.3

Mean:

\[ E(X)=0(0.2)+1(0.5)+2(0.3)=1.1 \]

Variance:

\[ \sum x^2p = 0 + 0.5 + 1.2 = 1.7 \] \[ \mathrm{Var}(X)=1.7-(1.1)^2 \]
\[ \mathrm{Var}(X)=0.49 \]
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