Vocab
Part 1
Part 1
Describe Discrete random variables
Discrete random variables take a countable set of possible values with gaps between them on the number line.
Mean
The average of numbers.
µ =np
Mean for binomial distribution
Determine whether the given random variable is a binomial random variable. If so, state its probability distribution. Genetics says that the genes children receive from their birth parents are independent from one child to another. Each child of a particular set of birth parents has probability 0.25 of having type O blood. Suppose these parents have 5 children. Let 𝑋 = the number of children with type O blood.
Binary? “Success”=has type O blood. “Failure”=doesn’t have type O blood.
Independent? Knowing one child’s blood type tells you nothing about another child’s blood type because they inherit genes independently from their birth parents.
Number? 𝑛=5
Same probability? 𝑝=0.25
This is a binomial setting, and 𝑋 is counting the number of successes (children with type O blood). So 𝑋 is a binomial random variable with n=5 and 𝑝=0.25.
Binomial Distribution: Finds the probability that exactly x successes occur during n trials where the probability of success on a given trial is equal to p.
binompdf(n, p, x)
Binomial Distribution
Counts successes in a fixed number of trials.
Transformation (Addition or Subtraction)
Changes the measures of center (mean, median), but not the shape or variability of the probability distribution.
µ = 1/p
Mean for geometric distribution
A professional soccer player succeeds in scoring a goal on 84% of her penalty kicks. Assume that the succes of each kick is independent. What is the probability that the first time she fails to score a goal is on her fifth penalty kick?
p(x=5)
1-.84=.16
geometpdf(.16,5)=.08
Geometric Distribution: Calculates the probability that the first success occurs precisely on the k-th trial.
geometpdf(p, x)
Geometric Distribution
Counts trials until first success.
Transformation (Multiplication or Division)
Changes the measures of center (mean, median) and variability (standard deviation, range, interquartile range), but not the shape of the probability distribution
σ = np(1 − p)
Standard Deviation for binomial distribution.
Pain score,
𝑦
1 2 3 4 5
Probability,
𝑃(Y) = 0.1 0.2 0.3 0.3 ??
Find P(Y) if y=5
P(Y) = .1
Binomial Distribution:Finds the probability that x successes or fewer occur during n trials where the probability of success on a given trial is equal to p.
binomcdf(n, p, x)
10% Condition
A sample size should be no more than 10% of the total population size to ensure that observations can be treated as approximately independent.
σ = √((1 - p) / p)
Standard Deviation for geometric distribution.
53 + 14
67
Geometric Distribution: Calculates the cumulative probability that the first success occurs at or before the k-th trial.
geometcdf(p, x)
Standard Deviation
How much a data set typically deviates from its average (mean).
µ = E( X ) = ∑ x Pi ( x )
Discrete random variable, X
what does 140/2-3 equal
67