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QBIO 305 > Chapter 3 Flashcards > Flashcards

Chapter 3 Flashcards

1
Q

What is a probability?

A

a numerical quantity that expresses the likelihood of an event

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2
Q

How is the probability of event E written?

A

Pr{E}

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3
Q

What is Pr{E} always between?

A

0 and 1 (inclusive)

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4
Q

What is the frequentist interpretation of Probability?

A

Pr{E} = (number of times event E occurs)/ (number of times chance operation is repeated)

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5
Q

What is Bayesian probability?

A

expresses someone’s belief that an event will happen

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6
Q

In a Bayesian analysis what is there before any data is collected?

A

-prior probability
-then when data is collected there is a calculation to update to a posterior probability

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7
Q

Know how to use probability trees

A
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8
Q

What is the hypothetical 1,000?

A

Multiply the probabilities in the tree by 1,000 and make a table

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9
Q

What is the probability of all possible events?

A

1

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10
Q

What is the probability that an event does not happen?

A

1-Pr{E} = E complement (E^c)

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11
Q

What is the addition rule for any two events E1 and E2?

A

Pr{E1 or E2} = Pr{E1} + Pr{E2} - Pr{E1 and E2}

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12
Q

When are two events E1 and E2 said to be disjoint?

A

Pr{E1 and E2} = 0 meaning no overlap

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13
Q

What is the conditional probability of E2 given E1?

A

Pr{E2|E1} = Pr{E2 and E1) / Pr{E1}

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14
Q

What is the multiplication rule for any two events E1 and E2?

A

Pr{E1 and E2} = Pr{E1} x Pr{E2|E1} = Pr{E2 and E1}
Also:
Pr{E1 or E2} = Pr{E2 or E1}

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15
Q

When are events E1 and E2 independent?

A

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

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16
Q

What is the Law of Total Probability?

A

For any two events E and F,
Pr{F} = P{E and F} + P{E^c and F}

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17
Q

Using the Multiplication Rule on the Law of Total Probability?

A

P{F} = P{F|E}P{E} + P{F|E^c}P{E^c}

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18
Q

What is Bayes’ Theorem?

A

P(E|F) = (P(F|E)P(E))/P(F)

19
Q

What do you get when you combine Bayes’ Theorem with the Law of Total Probability?

A

P(E|F) = (PF|E)P(E)/P(F|E)P(E) + P(F|E^c)p(E^c)

20
Q

Understand this slide

A

(Monte Hall Problem)

21
Q

What is a random varaible?

A

-a variable that takes on numerical values that depend on the outcome of a chance operation

22
Q

For a discrete random variable what is the sum of all histogram heights?

A

1

23
Q

For a continuous random variable what is the integral under the density curve?

A

1

24
Q

What is the Continuity Paradox?

A

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

25
Q

What is the mean of a random variable? And what is it also called?

A

Mean is also the expected value

26
Q

What is the variance (sigma)^2 of a random variable?

A
27
Q

What does the mean refer to relative to the random variable distribution?

A

the center

28
Q

What does the variance refer to in regards to the random variable distribution?

A

the spread or dispersion

29
Q

How do sample mean and random variable mean differ?

A

-the sample mean is different for different samples
-the random variable mean is aways the same

30
Q

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?

A

ux + uy

31
Q

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?

A

ux - uy

32
Q

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

a + X

33
Q

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?

A

b(ux)

34
Q

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?

A

b^2(sigma)^2x (a doesn’t matter)

35
Q

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?

A

Assuming they are independent:
var(x) + var(y)

36
Q

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?

A

Assuming they are independent:
var(x) + var(y)

37
Q

What is size bias and provide and example?

A

-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

38
Q

What is a Bernoulli random variable?

A

-a trial with probability p of success and probability (1-p) of failure
P(Y=1) = p
P(Y=0) = 1-p

39
Q

What is the expected value and variance of a Bernoulli random variable?

A

Ey = p
Var(y) = (1-p)p

40
Q

What is a binomial random varaible?

A

-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

41
Q

What is the expected value and variance of a binomial random variable?

A

Ez = np
Var(Z) = np(1-p)

42
Q

What is the formula for nCj or “n choose j”

A

n! / j!(n-j)!

43
Q

What is the binomial formula for n independent rails each w/ prob. p of success?

A
44
Q

What is the difference between efficacy and effectiveness?

A

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

QBIO 305 (6 decks)

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