Activity 8

MATH 216: Statistical Thinking

Discrete Random Variables Practice

Time Allocation: 15 minutes total

Part 1: Probability Distribution Construction (5 minutes)

Scenario: Tossing two coins, let x = number of heads

Context: When we toss two fair coins, we can get 0, 1, or 2 heads. This is a classic example of a discrete random variable where we need to identify all possible outcomes and their probabilities. Remember that for fair coins, each outcome (HH, HT, TH, TT) is equally likely.

Complete the probability distribution table:

x (Number of Heads) Possible Outcomes P(x)
0
1
2

Verification: Sum of all probabilities = _________________

Think About It: Why must the sum of all probabilities equal 1?

Part 2: Probability Distribution Analysis (5 minutes)

Given Distribution:

x -2 -1 0 1 2
p(x) .10 .15 .40 .30 .05

Context: This probability distribution shows the likelihood of different values for a discrete random variable X. Notice that all probabilities are between 0 and 1, and they sum to 1 (verify: 0.10 + 0.15 + 0.40 + 0.30 + 0.05 = 1.00).

Key Concepts to Remember:

  • \(P(X \leq a)\) means the probability that X takes a value less than or equal to a
  • \(P(X > a)\) means the probability that X takes a value greater than a
  • \(P(a \leq X \leq b)\) includes both endpoints a and b
  • \(P(a \leq X < b)\) includes a but excludes b

Calculate:

  1. \(P(X \leq 0)\) = _________________________

  2. \(P(X > -1)\) = ________________________

  3. \(P(-1 \leq X < 1)\) = ____________________

  4. \(P(-1 \leq X \leq 1)\) = ____________________

Part 3: Special Distributions (5 minutes)

Drought Research: \(p(x) = (0.3)(0.7)^{x-1}\) for x = 1, 2, 3, …

Context: This is a geometric distribution! It models the number of trials needed for the first success in a sequence of independent Bernoulli trials. Here, “success” is a drought year with probability 0.3, and we want to know how many years until the first drought occurs.

Key Properties of Geometric Distribution:

  • Models “waiting time” until first success
  • Probability of success on any trial = 0.3
  • Probability of failure on any trial = 0.7
  • Formula: \(P(X = x) = p(1-p)^{x-1}\) where p = probability of success

Calculate:

  1. \(P(\text{exactly 3 years until drought})\) = ____________________

  2. \(P(\text{no more than 2 years})\) = __________________________

Insurance Analysis:

Context: Calculate the average gain an insurance company makes per policy. The company gains the premium if the person lives, but loses money if the person dies.

Given Information:

  • Policy value: $10,000
  • Premium charged: $290
  • Probability of death: 0.001 (probability of survival = 0.999)

What the company gains in each scenario:

  • If person lives: company keeps the $290 premium → gain = $290
  • If person dies: company pays $10,000 but collected $290 → gain = $290 - $10,000 = -$9,710

Expected gain = ____________________________________

  • Show work: ________________________________________