STATS 10 Lecture Notes - Lecture 7: Binomial Distribution, Geometric Distribution, Standard Deviation
Chapter 6: Modeling Random Events: The Normal and Binomial Models
Probability distributions are models of random experiments
● Random: describes outcome of a scenario that canot be predicted w/ certainty
○ Event: outcome/set of outcomes from a random scenario
● Probability model: describes how we think data from a random scenario is generated→ represents a summary of
the unobservable data generating mechanism
○ We make an assumption that the probability model is a reasonably accurate description of how the data
is generated
○ We can tell if a model is good by how well the probabilities that the model predicts are matched by real-
life outcomes (just like the regression line)
● Distribution of a sample is a list that records
○ (1) the values that were observed in the data and
○ (2) the frequencies of these values
○ Distributions organize data to make comparisons and understand the variation in the data
● Probability distribution/probability distribution function (pdf) = description of the random scenario that tells
us
○ (1) the possible outcomes of the random scenario
○ (2) the probability of each outcome
● OF NUMERICAL VALUES, we can split the category into two subsections
○ Discrete outcomes/variables: numerical values that can be listed/counted
○ Continuous outcomes/variables: numerical values that occur over a range and cannot be
listed/counted
● Like how the way we visiualize data distribution depended on type of variable (numerical/categorical), the way we
describe/visualize a probability distribution will depend on the types of outcomes that are possible from a random
scenario (discrete or continuous)
○ Discrete: table/graph, can even
make a formula/equation
■ To make a graph for
discrete probability
distribution, we draw a
vertical line above the #
line where each outcome
value occurs, its height
proportional to each
outcome’s probability
■ Geometric distribution:
models the number of
independent trials until a
success
● E.g. probability of rolling a die x times until we see a 6 is
○ P(rolling x times) = (᪥)x-1 x (᪤)
○ For continuous probability distributions, we cannot make a table to list
all possible outcomes since they occur over a RANGE of values →
focus on finding probability of a range of outcomes rather than
considering individual outcomes
■ Probability density curve → curve above x-axis where the
area under the curve for a given range (interval) of x-values
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represents the probability that the outcome will be in that range
■ WARNING: the total area under the curve must be equal to 1! If not, the curve is not valid
● E.g. probability of wait time for coffee is 3-5 minutes
■ Probability density curve is usually a probability model → there is no way to observe enough
outcomes to have absolute certainty that the curve we use is the truth, so we make an
assumption that the probability model is a reasonable assumption of the underlying data
generating mechanism
● Many random phenomena have been shown to follow certain patterns that allow us to
have more confidence in certain settings
● continuous/uniform distribution: flat distribution shown below
○
The normal model
● Normal curve/distribution or bell curve/gaussian model: probability density curve that is symmetric and
unimodal
○ One of the most frequently used probability models in science
○ Features heaviliy in the Central Limit Theorem
○ Mean of a probability distribution= balancing point of a a probability distribution, denoted by mu (μ)
○ Standard deviation of a probability distribution = measures how far away the typical values are from
the mean, denoted by sigma (σ)
■ Do not confuse mean/standard dev of probability distributions w/ mean and standard deviations of
data (x¯ and s)
■ Mean and standard dev of probability distribution determines exact shape of normal distribution
● Normal distriution=perfect symmetric so mean=exact center
● wide/low normal curve has larger standard deviation (more area spread away from mean)
● narrow/tall normal curve has smaller standard deviation (more area clustered around
mean)
○ The notation N(μ, σ) represents a Normal distribution that is centered at the value μ (mean) and whose
spread is measured by the value σ (standard dev)
■ E.g. children’s heights, Normal model N(68, 2)
○ Since Normal distribution models continuous variables, area under the normal curve for a given range of
x-values represents the probability that the variable will be in that range
■ Probability of child will be taller than 65 inches is the same whether or not we include 65
inches → in continuous variables, strict inequality statements are interchangeable with non-strict
inequality statements
● This is NOT true for discrete variables
○ The formula for Normal curve and computing the area under the curve are both complicated! We use
either comp. Software or probability tables
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