Binomial Random Variables Study Notes

Notes on Binomial Random Variables

Introduction to Binomial Random Variables

  • When the same chance process is repeated several times, interest often lies in whether a particular outcome occurs on each repetition.

  • Examples:

    • ESP Test:

    • Chance Process: Choose a card at random (star, wave, cross, circle).

    • Outcome of Interest: Person identifies the card correctly.

    • Random Variable: $X = $ number of correct identifications.

    • Shipping Company Claim:

    • Chance Process: Randomly sample a shipment and check if it arrived on time.

    • Outcome of Interest: Shipment arrives on time.

    • Random Variable: $Y = $ number of on-time shipments.

    • Pass the Pigs Game:

    • Chance Process: Roll pig dice.

    • Outcome of Interest: Getting a "pig out" (both pigs land on opposite sides).

    • Random Variable: $T = $ number of rolls until pigs out.

  • Random Variables:

    • Some (like $X$ and $Y$) count occurrences of an outcome in fixed repetitions (binomial random variables).

    • Others (like $T$) count repetitions until the outcome occurs (geometric random variables).

I. Definition of Binomial Setting

  • A binomial setting arises from performing several independent trials of the same chance process and recording the number of times a particular outcome occurs.

  • Four Conditions for a Binomial Setting:

    1. Binary: Each trial’s outcomes can be classified as “success” or “failure.”

    2. Independent: Trials must be independent; the result of one trial does not affect another.

    3. Number: Number of trials $n$ must be fixed in advance.

    4. Success: There is a constant probability $p$ of success on each trial.

II. Binomial Random Variable

  • If all binomial setting criteria (BINS) are met, then $X$ is a binomial random variable.

III. Binomial Distribution

  • The probability distribution of $X$ is a binomial distribution with parameters $n$ and $p$.

  • Parameters:

    • $n$: Number of trials.

    • $p$: Probability of success on any one trial.

  • Possible values of $X$: Whole numbers from 0 to $n$.

IV. Notation

  • $n$: Number of trials.

  • $p$: Probability of success.

  • $X ilde{B}(n, p)$: Denotes $X$ as a binomial random variable with $n$ trials and probability $p$ of success.

  • In a binomial setting, $X$ represents the number of successes in $n$ independent trials.

V. Examples of Binomial Distribution

  • Example 1: Blood Types and Aces

    • A. Genetics scenario:

    • Random Variable: $X = $ number of children with type O blood.

    • Conditions:

      • Binary: Success = type O, Failure = not type O.

      • Independent: Yes, blood types are inherited independently.

      • Number: $n = 5$.

      • Success Probability: $p = 0.25$.

    • Conclusion: Binomial Setting - $X ilde{B}(5, 0.25)$.

    • B. Shuffling cards:

    • Random Variable: $Y = $ number of aces observed in the first 10 turns.

    • Condition: Trials are not independent (cards are not replaced).

    • Conclusion: Not a binomial distribution.

    • C. Drawing cards with replacement:

    • Random Variable: $W = $ number of draws required to get an ace.

    • Conclusion: Not a binomial distribution because number of trials is not fixed.

VI. Inheriting Blood Type

  • Scenario: Parents with children having a probability of type O blood of 0.25.

  • Random Variable: $X = $ number of children with type O blood.

  • Binomial parameters: $X ilde{B}(5, 0.25)$.

    • Calculate probabilities:

    • What’s $P(X = 0)$?

      • Probability that none of the 5 children have type O blood is computed using the multiplication rule for independent events:

      • $P(X = 0) = (0.75)^5 = 0.2373$.

    • What’s $P(X = 1)$?

      • Explore all ways to get exactly 1 child with type O blood and compute:

      • $P(X = 1) = 5 imes (0.25)(0.75)^4 = 0.39551$.

VII. Finding $P(X = 2)$

  • To find $P(X = 2)$:

    • Compute one arrangement (SFSFF), which uses the multiplication rule:

    • $P(X = 2) = 10 imes (0.25)^2 imes (0.75)^3 = 0.26367$.

VIII. Binomial Probability Formula

  • If $X$ follows the binomial distribution, the formula is:

    • P(X=k)=(nk)pk(1−p)n−kP(X = k) = \binom{n}{k} p^k (1 - p)^{n-k}

  • Notation:

    • $inom{n}{k}$: Binomial Coefficient - represents how many ways $k$ successes can occur in $n$ trials,

    • Formula: (nk)=racn!k!(n−k)!\binom{n}{k} = rac{n!}{k!(n-k)!}.

IX. Example 4: Finding Probabilities with Binomial Distribution

  • Probability that exactly 3 of the children have type O blood:

    • $P(X = 3) = inom{5}{3} (0.25)^3(0.75)^2 = 0.0879$.

  • Probability of having more than 3 children with type O blood:

    • $P(X > 3) = P(X = 4) + P(X = 5)$ leads to $P(X > 3) = 0.01563$.

X. Calculator Commands for Binomial Probability

  • $binompdf(n, p, k)$ computes $P(X = k)$.

  • $binomcdf(n, p, k)$ computes $P(X ≤ k)$.

  • For $P(X ≥ k)$ and $P(X > k)$, transform them into:

    • $P(X ≥ k) = 1 - P(X ≤ k - 1)$ and $P(X > k) = 1 - P(X ≤ k)$.

XI. Example 5 - Application of Binomial Probability

  • For $X ilde B(100, 0.3)$, find:

    • $P(X ≤ 25)$ using calculator command: $binomcdf(trials: 100, p: 0.3, X value: 25)$ = 0.163.

XII. Mean and Standard Deviation of Binomial Random Variable

  • If $X$ has a binomial distribution,

    • Mean: ext{Mean}(bc_x) = np

    • Standard Deviation: ext{Standard Deviation}(c3_x) = ext{sqrt}(np(1-p)).

XIII. Conclusion: Understanding Binomial Distributions

  • Random variables in binomial settings follow specific distributions.

  • Recognizing success/failure outcomes and independence is crucial in determining if a binomial distribution applies. Understanding the associated probabilities, means, and standard deviations is essential for deeper statistical analysis.