Discrete RV

Discrete Distributions

  • Focuses on discrete random variables.

Probability Mass Function (PMF)

  • Defines probabilities for distinct events.

Properties of Random Variables

  • Location: Expectation (average value).

  • Spread: Variance (average squared deviation from the mean).

Expectation of Discrete Random Variables

  • Calculated as E(X)=∑<em>i=1nx</em>iP(X=xi)E(X) = \sum<em>{i=1}^{n} x</em>i P(X = x_i).

Weighted Dice Example

  • Example calculation shows expected value for weighted 6-sided dice.

Discrete PMF: Bernoulli Distribution

  • Outcomes: two possible results with respective probabilities.

  • HPV prevalence (43% vs 57%) example given.

Probability Calculations for HPV

  • For 2 people entering a room, probabilities for different outcomes (0, 1, 2 with HPV).

  • For 3 people, similar calculations extend further.

Counting Combinations

  • Combinations for outcomes calculated using factorial notation: C(n,x)=n!x!(n−x)!C(n, x) = \frac{n!}{x!(n-x)!}.

Binomial Distribution

  • Counts successes in nn Bernoulli trials; trials are independent; constant probability pp for success.

  • Not a binomial experiment if nn is not fixed.

Binomial PMF

  • PMF represented as P(X=x)=C(n,x)px(1−p)n−xP(X = x) = C(n, x) p^x (1-p)^{n-x}.

Hypertension Example

  • Prevalence of hypertension (29%); probability of finding 3 out of 20 cases calculated.

Expected Value and Variance

  • Expected value for hypertension cases: E(X)=npE(X) = np.

  • Variance formula: Var(X)=np(1−p)Var(X) = np(1-p).

  • For hypertension example, expected cases are 5.8, variance is 4.1.

Exercises

  • Included exercises on normal distributions, Z-scores, and GRE score interpretations.