Lecture 5: Probability: Binomial Distribution, Normal Distribution, standard normal distribution, and Z-Tables

Independence

Two events, A and B, are independent if P(A | B) = P(A) or if P(B | A) = P(B)

To calculate probability…

• Typically, count the number of participants that had the characteristic of interest and divide by the population size

• For conditional probabilities, the population size (denominator) was modified to reflect the subpopulation of interest

• P(A and B) = P(A) x P(B) if A and B happen simultaneously

• P (C and D) = P(C) + P(D) if C and D are mutually exclusive

Binomial distribution

• Model for dichotomous outcome

• The binomial formula generates the probability of observing exactly x successes out of n

• We typically do not talk about mean and variance for dichotomous variables, but we can quantify mean and variance for every probability distribution

• Mean and variance of (random variables generated from) the binomial distribution

Binomial distribution: Model for dichotomous outcome

1. Two possible values (responses) for each data point: success and failure

2. Replications of the process are independent

3. P(success) is constant for each replication

Binomial distribution: The binomial formula generates the probability of observing exactly x successes out of n

P(x success) = (n! / x! (n-x)!) p^x (1-p)^n-x

Binomial Distribution: Notation

• n = number of times the process is repeated

• p = P(succes) where success is outcome of interest

• x = number of successes of interest 0 ≤ x ≤ n

• ! = factorial; k! = k(k-1)(k-2)… 1

What the binomial formula does in one step

The binomial coefficient (the fraction with factorials) figures out how many such orderings are possible and then multiply by the common probability

Binomial distribution: Mean and variance of (random variables generated from) the binomial distribution

• Mean or expected number of successes: μ = np

• Variance: σ² = np(1-p)

• Standard deviation: σ

Normal distribution

• Aka Gaussian distribution

• Model for a continuous outcome when the distribution is well described by a bell-shaped curve

• Notation: μ = distribution mean and σ = distribution standard deviation

• x-axis is used to display the scale of the characteristic/variable being analyzed (e.g., height, weight, systolic blood pressure)

• y-axis reflects the probability density (relative likelihood) of observing each value

• Curve is highest in the middle, suggesting that the values near the middle have higher probabilities or are more likely to occur; values at either extreme are much less likely to occur

• Mean = median = mode (hump/most frequent value)

• Area under the density curve before X=a represent P(X ≤ a)

Properties of normal distribution

1. The normal distribution is symmetric about the mean i.e. P(X > μ) = P(X < μ) = 0.5.

   1. The mean = the median = the mode

2. The mean and variance, μ and σ², completely characterize the normal distribution

3. P(a < X < b) = the area under the normal density curve from a to b

   1. 𝑃 (𝜇 − 𝜎 < 𝑋 < 𝜇 + 𝜎) = 0.68
      𝑃 (𝜇 − 2𝜎 < 𝑋 < 𝜇 + 2𝜎) = 0.95
      𝑃 (𝜇 − 3𝜎 < 𝑋 < 𝜇 + 3𝜎) = 0.99

4. The probability density function of a normal distribution is given by p(x) = (1 / σ√2π) e^{-(x-μ)² / 2𝜎²}

Standard normal distribution Z

• Normal distribution with μ = 0 and 𝜎 = 1

• P(-1 < X < 1) = 0.68

• P(-2 < X < 2) = 0.95

• P(-3 < X < 3) = 0.99

• Z = x-μ / 𝜎

Normal distribution calculate probability

• Template: Computing probabilities about normal distributions

• For the normal distribution, and for other distributions for any continuous variable, there is no area in a single line, and thus the absolute likelihood P(x = specific value) is defined as 0

Normal distribution calculate probability: Template: Computing probabilities about normal distributions

• First standardize or convert a problem about a normal distribution (X) into a problem about the standard normal distribution (Z)

• Then use the Z table to compute the desired probability

Normal distribution calculate probability: For the normal distribution, and for other distributions for any continuous variable, there is no area in a single line, and thus the absolute likelihood P(x = specific value) is defined as 0

This is not true for the binomial distribution and for other probability distributions for discrete/categorical/ordinal variables

Percentiles of the normal distribution

• The k^th percentile is defined as the score that holds k percent of the scores below it

• 90^th percentile is the score that holds 90% of the scores below it

Percentiles of the normal distribution: 90^th percentile is the score that holds 90% of the scores below it

• Q₁ = 25^th percentile

• Median = 50^th percentile

• Q₃ = 75^th percentile

For the normal distribution, the following is used to compute percentiles

x = μ + z 𝜎

z = x - μ / 𝜎

where

μ = mean of the random variable X

𝜎 = standard deviation

z = value from the standard distribution for the desired percentile