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Notes › EENG 3421: Probability and Statistics for Engineers Lecture 21

Conditional Expected Values

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Conditional Probability of Sets Under a Conditional PMF/PDF

Theorem 21.1 (Properties of Conditional PMF and Conditional PDF).

Discrete

  • $P_{X\mid B}(x) \ge 0$ for $x \in B$
  • $\sum_{x \in B} P_{X\mid B}(x) = 1$
  • for any set $C$
$$ P[C \mid B] = \sum_{x \in C} P_{X\mid B}(x) $$

Continuous

  • $f_{X\mid B}(x) \ge 0$ for $x \in B$
  • $\int_{B} f_{X\mid B}(x)\,dx = 1$
  • for any set $C$
$$ P[C \mid B] = \int_{C} f_{X\mid B}(x)\,dx $$

Conditional Expected Value

Definition 21.2 (Conditional Expected Value).

The conditional expected value of random variable $X$ given condition $B$ is

Discrete

$$ E[X \mid B] = \sum_{x \in B} x\,P_{X\mid B}(x) $$

Continuous

$$ E[X \mid B] = \int_{-\infty}^{\infty} x\,f_{X\mid B}(x)\,dx $$

Total Expectation with a Partition

Theorem 21.3 (Total Expectation over a Partition).

For a random variable $X$ resulting from an experiment with partition $B_1,\dots,B_m$

$$ E[X] = \sum_{i=1}^{m} E[X \mid B_i]\,P[B_i] $$

Conditional Expectation of a Function

Theorem 21.4 (Conditional Expected Value of a Function).

The conditional expected value of $Y=g(X)$ given condition $B$ is

Discrete

$$ E[Y \mid B] = E[g(X)\mid B] = \sum_{x \in B} g(x)\,P_{X\mid B}(x) $$

Continuous

$$ E[Y \mid B] = E[g(X)\mid B] = \int_{-\infty}^{\infty} g(x)\,f_{X\mid B}(x)\,dx $$

Conditional Variance and Standard Deviation

Definition 21.5 (Conditional Variance and Standard Deviation).

The conditional variance of $X$ given event $B$ is

$$ \mathrm{Var}[X \mid B] = E[(X-\mu_{X\mid B})^2 \mid B] = E[X^2 \mid B] - \mu_{X\mid B}^2 $$

The conditional standard deviation is

$$ \sigma_{X\mid B} = \sqrt{\mathrm{Var}[X \mid B]} $$

References

Sources

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