Midterm 2: 2.1-2.2 Flashcards
What is a random experiment?
A process that leads to a single outcome which cannot be predicted with certainty.
What is an element in probability?
A specific outcome of an experiment.
What is the sample space denoted as S?
The collection of all possible outcomes for a given experiment.
Define an event in probability.
A collection of elements that are of interest or share a common characteristic in a given experiment.
What does P(A) represent?
The probability of event A occurring when an experiment is performed.
What does the complement of event A, denoted as A’, represent?
The set of all elements in the sample space S that do not belong to A.
Probability of X
What is the formula for the complement of event A?
P(A) + P(A’) = 1.
What is the union of events A and B denoted as?
A ∪ B.
What does the union of A and B represent? + formula
The set of all elements that belong to A or B or both.
What is the formula for the intersection of events A and B?
P(A ∩ B) = P(A) + P(B) - P(A ∪ B).
What is mutual exclusivity in probability?
Events A and B are mutually exclusive if there are no elements that are in both A and B.
What is conditional probability?
A probability that reflects additional knowledge that may affect the outcome of an experiment.
How is conditional probability denoted?
P(A|B) represents the probability of A given B.
What is the formula for calculating conditional probability?
P(A|B) = P(A ∩ B) / P(B).
What is the law of total probability?
P(A) = P(A ∩ B) + P(A ∩B ^c).
What is a contingency table?
A table that organizes the probability relationships between two events A and B.
What does it mean for two events A and B to be independent?
The occurrence of A does not affect the probability of B, and vice-versa.
What is one way to prove independence between events A and B?
If P(A|B) = P(A) and P(B|A) = P(B), then A and B are independent.
Fill in the blank: The probability of event A is conditional upon another event occurring (event B), so now, P(A|B) is the ratio of _______.
P(A ∩ B) over P(B).
What is DeMorgan’s law in probability?
P(A’ ∩ B’) = P(A ∪ B)’ = 1 - P(A ∪ B).
True or False: If events A and B are mutually exclusive, then P(A ∩ B) > 0.
False.
What keyword indicates conditional probability?
If, given, assuming, knowing.
Define event A as ‘the sum of the numbers is odd.’ What is u(A)?
u(A) represents the number of elements in event A.
What is the probability that a randomly selected person will test ‘negative’ for the disease?
Calculated based on the constructed contingency table.
What is the probability that a randomly selected person will have the disease given that they test negative for the disease?
Calculated using conditional probability.
What is a random variable?
A characteristic, attribute, or outcome that can vary from observation to observation.
What is the purpose of a probability distribution table?
To display all the values a discrete random variable can assume along with the associated probabilities.
What are the key summary values used to describe a random variable’s behavior?
Expected value, variance, and standard deviation.
What is the expected value of a random variable A?
The average value of A over an infinite number of repetitions of an experiment.
How is variance defined for a discrete random variable A?
It is a measure of the variability of A over an infinite number of repetitions of an experiment.
What is the standard deviation of a discrete random variable?
The positive square root of the variance.
What defines a binomial random variable?
An experiment with a fixed number of identical, independent trials with the same probability of success.
What are the parameters needed to compute probabilities for a binomial random variable?
- n — the number of trials in the experiment
- p — the probability of success
What is the probability mass function (pmf) for a binomial random variable?
The equation used to find the probability that a random variable assumes a particular value x.
What is the R function to find P(X = x) for a binomial random variable?
dbinom(x, n, p)
What is a continuous random variable?
A variable that can take on an infinite number of values.
What is the probability density function (pdf)?
The function that defines the curve representing the distribution of a continuous random variable.
What is the probability that a continuous random variable equals a specific value?
It is equal to zero.
What does the area under the curve of a pdf represent?
It represents the probabilities for intervals of values.
What is the normal distribution often assumed for?
Many variables in the natural and social sciences when their exact distributions are not known.
True or False: Discrete random variables can take on any value within a range.
False
Fill in the blank: A probability distribution table is appropriate for _______.
discrete random variables.
What is the notation for a binomial random variable?
X ~ Binomial(n, p)
What does the notation P(A) represent?
The probability of event A occurring.
What is the formula for calculating the expected value of a binomial random variable?
E(X) = n * p
What is the formula for calculating the variance of a binomial random variable?
VAR(X) = n * p * (1 - p)
What does standardizing a normal variable involve?
Calculating its z-score.
What does a probability distribution graph help to observe?
The shape of the distribution and the behavior of a random variable.
What is the normal (or Gaussian) distribution often assumed for?
A wide variety of variables in the natural and social sciences when exact distributions are not known.
Examples include scores on the Stanford-Binet IQ test, adult heights, and gestation time of elephants.
What two parameters are needed to compute probabilities for a normal random variable?
- Mean (y)
- Standard deviation (σ)
How is the notation for a normal distribution expressed?
X ~ N(μ, σ) where μ is the mean and σ is the standard deviation.
What is the shape of normal distributions?
Bell-shaped and symmetric about the mean.
What does the Probability Density Function (pdf) define?
The shape of a normal random variable.
What does the Cumulative Density Function (cdf) allow us to do?
Define and calculate probabilities for a normal random variable.
What R function is used to find P(X ≤ x)?
pnorm(x, mean, sd)
What R function is used to find a specific value of X given a known probability?
qnorm(p, mean, sd)
What are the expected value and variance for a normal random variable Z?
- E(X) = μ
- VAR(X) = σ²
What is the normal distribution used to model in the context of the Lilac Bloomsday Run?
The finishing time of participants.
How do you calculate the proportion of participants finishing in under 90 minutes?
Using the R function pnorm(x, mean, sd)
What is the maximum finishing time for the bottom 20% of participants?
Calculated using the R function qnorm(p, mean, sd)
What is the standard normal distribution characterized by?
Mean of 0 and standard deviation of 1.
What are the three main purposes of the standard normal distribution?
- Ease of calculating probabilities
- Use of standardized scores (z-scores)
- Use in statistical inference
What does a z-score represent?
How many standard deviations a value is from the mean.
What does the empirical rule state about a normal distribution?
- Approximately 68% of observations within +/- 1 standard deviation
- Approximately 95% within +/- 2 standard deviations
- Approximately 99.7% within +/- 3 standard deviations
What is another name for the empirical rule?
The 68-95-99.7 rule.
What were standard normal tables developed for?
To compute probabilities without the need for calculus.
What is the formula for calculating the z-score?
z = (X - μ) / σ
How can a z-score be calculated in R?
Using the scale function: scale(x, center, scale)
What is the mean and standard deviation of the Stanford-Binet IQ test?
- Mean = 100
- Standard deviation = 15
How do you standardize a score of 106 on the IQ test?
Using the formula z = (X - μ) / σ or the R function scale(x, center, scale)
What does a z-score of 3 indicate?
The corresponding x value is 3 standard deviations above the mean.
What does a z-score of -0.5 indicate?
The corresponding x value is half a standard deviation below the mean.
What does a z-score of 0 represent?
A value equal to the mean.
