Chapter 3 Flashcards
What is a probability?
a numerical quantity that expresses the likelihood of an event
How is the probability of event E written?
Pr{E}
What is Pr{E} always between?
0 and 1 (inclusive)
What is the frequentist interpretation of Probability?
Pr{E} = (number of times event E occurs)/ (number of times chance operation is repeated)
What is Bayesian probability?
expresses someone’s belief that an event will happen
In a Bayesian analysis what is there before any data is collected?
-prior probability
-then when data is collected there is a calculation to update to a posterior probability
Know how to use probability trees
What is the hypothetical 1,000?
Multiply the probabilities in the tree by 1,000 and make a table
What is the probability of all possible events?
1
What is the probability that an event does not happen?
1-Pr{E} = E complement (E^c)
What is the addition rule for any two events E1 and E2?
Pr{E1 or E2} = Pr{E1} + Pr{E2} - Pr{E1 and E2}
When are two events E1 and E2 said to be disjoint?
Pr{E1 and E2} = 0 meaning no overlap
What is the conditional probability of E2 given E1?
Pr{E2|E1} = Pr{E2 and E1) / Pr{E1}
What is the multiplication rule for any two events E1 and E2?
Pr{E1 and E2} = Pr{E1} x Pr{E2|E1} = Pr{E2 and E1}
Also:
Pr{E1 or E2} = Pr{E2 or E1}
When are events E1 and E2 independent?
If Pr{E1 and E2} = Pr{E1} x Pr{E2}
-Pr{E2|E1} = Pr{E2}
-Pr{E1|E2} = Pr{E1}
-two events are independent if knowing that one event occurred does not change the probability of the other event occurring
What is the Law of Total Probability?
For any two events E and F,
Pr{F} = P{E and F} + P{E^c and F}
Using the Multiplication Rule on the Law of Total Probability?
P{F} = P{F|E}P{E} + P{F|E^c}P{E^c}
What is Bayes’ Theorem?
P(E|F) = (P(F|E)P(E))/P(F)
What do you get when you combine Bayes’ Theorem with the Law of Total Probability?
P(E|F) = (PF|E)P(E)/P(F|E)P(E) + P(F|E^c)p(E^c)
Understand this slide
(Monte Hall Problem)
What is a random varaible?
-a variable that takes on numerical values that depend on the outcome of a chance operation
For a discrete random variable what is the sum of all histogram heights?
1
For a continuous random variable what is the integral under the density curve?
1
What is the Continuity Paradox?
That if you look at a density curve and you try to find the probability of some specific exact value instead of a range you get zero
What is the mean of a random variable? And what is it also called?
Mean is also the expected value
What is the variance (sigma)^2 of a random variable?
What does the mean refer to relative to the random variable distribution?
the center
What does the variance refer to in regards to the random variable distribution?
the spread or dispersion
How do sample mean and random variable mean differ?
-the sample mean is different for different samples
-the random variable mean is aways the same
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
What is the mean of X + Y?
ux + uy
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
What is the mean of X - Y?
ux - uy
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
Let a and b be constants.
What is the mean of a + X?
a + X
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
Let a and b be constants.
What is the mean of bX?
b(ux)
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
Let a and b be constants.
What is the variance of a + bX?
b^2(sigma)^2x (a doesn’t matter)
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
Let a and b be constants.
What is the variance of X + Y?
Assuming they are independent:
var(x) + var(y)
Consider the random variables X and Y where X has a mean of ux and variance of (sigma)^2x while Y has a mean uy and a variance of (sigma)^2y.
Let a and b be constants.
What is the variance of X - Y?
Assuming they are independent:
var(x) + var(y)
What is size bias and provide and example?
-when you judge an event to be more likely to occur simply because a particular category is larger
Example: you ask fathers how many children they have (1+99)/2 = 50
you ask children how many siblings they have including themselves (1+99x99)/100 = 98.02
What is a Bernoulli random variable?
-a trial with probability p of success and probability (1-p) of failure
P(Y=1) = p
P(Y=0) = 1-p
What is the expected value and variance of a Bernoulli random variable?
Ey = p
Var(y) = (1-p)p
What is a binomial random varaible?
-if there are n independent trial each with probability p of success count the number of successes
-a binomial random variable is the sum of n independent Bernoulli random variables
What is the expected value and variance of a binomial random variable?
Ez = np
Var(Z) = np(1-p)
What is the formula for nCj or “n choose j”
n! / j!(n-j)!
What is the binomial formula for n independent rails each w/ prob. p of success?
What is the difference between efficacy and effectiveness?
effectiveness - the ability of the vaccine to prevent outcomes in the “real world”
Efficacy is NOT a/an:
-observational study
-not just volunteer, don’t exclude people with co-morbidities
-the effectiveness will be lower then the efficacy
