Notes: Page-by-Page Discrete Random Variables and Moments

Page 1: Expected value basics and Bernoulli/geometric setup

• Expected value definition: $\mathbb{E}[X] = \sum_x x \cdot P(X=x)$

• Bernoulli example (two outcomes 0 or 1): if $P(X=1)=p<em>1$ and $P(X=0)=p</em>0=1-p<em>1$, then $\mathbb{E}[X] = 0\cdot p</em>0 + 1\cdot p<em>1 = p</em>1$

• Geometric setup (first success on trial X):

  • PMF: $P(X=x) = (1-p)^{x-1} p, \quad x=1,2,3,…$

  • Intuition: number of trials until first success; larger p → smaller expected number of trials.

Page 2

• Expected value of geometric distribution via calculus:

  • $\mathbb{E}[X] = \sum_{x=1}^{\infty} x (1-p)^{x-1} p$

  • Trick: differentiate the geometric series with respect to the parameter and interchange sum and derivative (under regularity conditions: differentiability and uniform convergence).

  • Result: $\mathbb{E}[X] = \frac{1}{p}$

  • Quick intuition: if p=0.2, then $\mathbb{E}[X] = 5$.

• Note on interchangeability: requires differentiability of the term and a justification (uniformly bounded sum); these are standard calculus conditions.

• Reminder: geometric mean/expectation is not trivial despite its simple form.

Page 3

• Not all distributions have a finite expected value.

  • Example: Cauchy distribution (heavy tails) can yield $\mathbb{E}[X]$ that does not exist (diverges).

  • If you try $\mathbb{E}[X] = \sum_x x \cdot P(X=x)$ and the sum diverges, the expectation does not exist.

• Practical takeaway: to guarantee an expectation exists, it is common to require \mathbb{E}[|X|] < \infty (absolute integrability).

• Relevance: heavy-tailed distributions have large tail probability mass which can make the expectation undefined or infinite.

Page 4

• Expectation of a function of a random variable: $\mathbb{E}[h(X)] = \sum_x h(x) \cdot P(X=x)$

• Profit example (three computers):

  • Cost: $3\times 500 = 1500$

  • Sell price: $1000$ per unit; unsold units repurchased at $200$ each.

  • If sold x units, revenue: $1000x + 200(3-x) = 800x + 600$

  • Profit: $\text{Profit} = \text{Revenue} - \text{Cost} = (800x + 600) - 1500 = 800x - 900$

  • Therefore: $\mathbb{E}[ ext{Profit}] = 800\mathbb{E}[X] - 900$

• If the function is linear, $h(x)=a x + b$, then $\mathbb{E}[h(X)] = a\mathbb{E}[X] + b$ (linearity of expectation).

• Practical use: compute (\mathbb{E}[X]) once and apply it to the linear form; for non-linear h, compute the full sum.

Page 5

• Variance definition for a random variable: $\operatorname{Var}(X) = \mathbb{E}[(X - \mu)^2],\quad \mu = \mathbb{E}[X]$

• Discrete distribution variance: $\operatorname{Var}(X) = \sum_x (x - \mu)^2 \cdot P(X=x)$

• Relation to data variance: distribution variance uses probability weights (not dividing by n or n-1); data variance uses sample-based averaging with a denominator (n-1) for unbiased estimation.

• Alternative form: $\operatorname{Var}(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2$ which is often easier to compute.

• Standard deviation: $\operatorname{SD}(X) = \sqrt{\operatorname{Var}(X)}$; interprets dispersion in the same units as X.

• Interpretation: larger variance/SD implies greater dispersion or volatility in the distribution.

Page 6

• Summary and preview:

  • We covered means (expected value) and variance for discrete random variables, including linear and non-linear transformations.

  • Noted that some distributions do not have finite expectations; absolute integrability is a typical condition for existence.

  • Next steps: discrete random variables chapter continuation, well-known discrete distributions, and when to use them to model real-life systems; practice computing means/variances and understanding distribution shapes.