Day 9

Math 216: Statistical Thinking

Bastola

Introduction to Binomial Random Variables

• Context of Binomial Experiments:

  • Used to model situations with two distinct outcomes.
  • Example: Flipping a coin results in either a Head (Success, \(S\)) or Tail (Failure, \(F\)).

Two Possible Outcomes:

• Success \((S)\)
• Failure \((F)\)

Consistent Probability:

• Probability of success \((p)\)
• Probability of failure \((q = 1 - p)\)

Characteristics of Binomial Experiments

• Essential Features:

  • Fixed Number of Trials \((n)\): Each experiment consists of \(n\) identical and independent trials.
  • Stable Probabilities: The probability of success \((p)\) and failure \((q = 1 - p)\) remains constant across trials.

Key Points:

• Independence of Trials: Each trial’s outcome does not influence another.
• Identical Trials: All trials follow the same probability model.

The Binomial Random Variable

• Defining the Variable:
  • A binomial random variable \(X\) counts the number of successes in \(n\) trials.
  • Notation: \(X \sim \operatorname{Bin}(n, p)\), where \(p\) is the probability of a success on any given trial.

Understanding \(X\):

• Represents the sum of successes in the experiment.
• Can take values from 0 (no success) to \(n\) (all successes).

The Binomial Probability Formula

• Calculating Probabilities:
  • The probability of exactly \(x\) successes in \(n\) trials is given by the binomial formula:

\[ P(x) = \binom{n}{x} p^x q^{n-x}, \quad x = 0, 1, 2, \ldots, n \]

Components:

• \(\binom{n}{x}\): The number of ways to choose \(x\) successes from \(n\) trials.
• \(p^x\): The probability of \(x\) successes.
• \(q^{n-x}\): The probability of \(n-x\) failures.

Refer to Activity 3. Use the formula for a binomial random variable to find the probability distribution of \(x\), where \(x\) is the number of adults who pass the fitness test.

\[\begin{align*} \begin{aligned} & p(0)=\frac{4!}{0!(4-0)!}(.1)^0(.9)^{4-0}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{(1)(4 \cdot 3 \cdot 2 \cdot 1)}(.1)^0(.9)^4=1(.1)^0(.9)^4=.6561 \\ & p(1)=\frac{4!}{1!(4-1)!}(.1)^1(.9)^{4-1}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{(1)(3 \cdot 2 \cdot 1)}(.1)^1(.9)^3=4(.1)(.9)^3=.2916 \\ & p(2)=\frac{4!}{2!(4-2)!}(.1)^2(.9)^{4-2}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{(2 \cdot 1)(2 \cdot 1)}(.1)^2(.9)^2=6(.1)^2(.9)^2=.0486 \\ & p(3)=\frac{4!}{3!(4-3)!}(.1)^3(.9)^{4-3}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{(3 \cdot 2 \cdot 1)(1)}(.1)^3(.9)^1=4(.1)^3(.9)=.0036 \\ & p(4)=\frac{4!}{4!(4-4)!}(.1)^4(.9)^{4-4}=\frac{4 \cdot 3 \cdot 2 \cdot 1}{(4 \cdot 3 \cdot 2 \cdot 1)(1)}(.1)^4(.9)^0=1(.1)^4(.9)=.0001 \end{aligned} \end{align*}\]

Based on the above result, what is \(P(X<3), P(X \geq 2), P(1 \leq X<3)\)?

Observations on Binomial Distributions

1. Unimodal Distribution:
   • Regardless of the value of \(p\), the distribution exhibits a single peak.
2. Skewness Based on ( p ):
   • Right Skew (p < 0.5): The distribution skews to the right, indicating more low-value outcomes.
   • Symmetry (p = 0.5): The distribution is symmetric, with equal probabilities on both sides of the mean.
   • Left Skew (p > 0.5): The distribution skews to the left, indicating more high-value outcomes.

As \(n\) increases, the binomial distribution approximates a bell-shaped curve.

Statistics of Binomial Distribution

Statistical Properties:

• Mean (\(\mu\)): \(\mu = np\)
• Variance (\(\sigma^2\)): \(\sigma^2 = np(1-p)\)
• Standard Deviation (\(\sigma\)): \(\sigma = \sqrt{np(1-p)}\)

Insights:

• The mean indicates the expected number of successes.
• Variance and standard deviation provide insight into the spread of the distribution around the mean.