Notes › EENG 3421: Probability and Statistics for Engineers Lecture 10
Averages and Expected Value
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Table of Contents
Averages
- Different notions of average capture different ideas of a typical value
- Mean uses all data values and their magnitudes
- Median is based on ordering only
- Mode is based on frequency only
Definition 10.1 (Mode).
A mode of random variable $X$ is a number $x_{\text{mod}}$ satisfying
$$ P_X(x_{\text{mod}}) \ge P_X(x) \quad \text{for all } x $$Definition 10.2 (Median).
A median, $x_{\text{med}}$, of random variable $X$ is a number that satisfies
$$ P[X \le x_{\text{med}}] \ge \frac12 $$$$ P[X \ge x_{\text{med}}] \ge \frac12 $$Expected Value
- Expected value is the probability weighted average of the possible values of a random variable
- For discrete $X$, the weights are given by the PMF
- Expected value is also called the mean of $X$
- Notation
- $E[X]$
- $\mu_X$
Definition 10.3 (Expected Value).
The expected value of $X$ is
$$ E[X] = \mu_X = \sum_{x \in S_X} x P_X(x) $$- Connection to the empirical average
- If a value $x$ appears $N(x)$ times in $n$ observations, then the sample mean is $$ \frac{1}{n}\sum_{\text{obs}} x $$
- Grouping by values gives $$ \frac{1}{n}\sum_x x N(x) = \sum_x x \frac{N(x)}{n} $$
- $\frac{N(x)}{n}$ is the relative frequency of $x$
- In the limit, relative frequency approaches $P_X(x)$
- This motivates the expected value formula as a limiting average
Proposition 10.4 (Expected Value as a Weighted Average).
If $S_X = \{x_1,x_2,\dots\}$ then
$$ E[X] = \sum_i x_i P[X=x_i] $$where the weights $P[X=x_i]$ are nonnegative and sum to $1$
Expected Value for Common Families
Theorem 10.5 (Expected Value of Bernoulli $(p)$).
If $X$ is Bernoulli $(p)$ then
$$ E[X] = p $$Proof. For Bernoulli $(p)$
$$ P_X(0)=1-p,\quad P_X(1)=p $$So
$$ E[X] = 0\cdot(1-p) + 1\cdot p = p $$
Theorem 10.6 (Expected Value of Geometric $(p)$).
If $X$ is Geometric $(p)$ then
$$ E[X] = \frac{1}{p} $$Theorem 10.7 (Expected Value of Poisson $(\alpha)$).
If $X$ is Poisson $(\alpha)$ then
$$ E[X] = \alpha $$Theorem 10.8 (Expected Values of Other Discrete Families).
- If $X$ is Binomial $(n,p)$
- If $X$ is Pascal $(k,p)$
- If $X$ is Discrete Uniform $(k,\ell)$
References
- Course lecture slides 15
Sources
- Course lecture slides 15