Week 5 SCM (resampling and simulation)

Question 1

what are the 4 steps to running a monte carlo simulation

Answer 1

Define a domain of possible values
Generate random numbers within that domain from a probability distribution
Perform a computation using the random numbers
Combine the results across many repetitions

Question 2

what does randomness mean in statistics

Answer 2

unpredictable

Question 3

what is randomness in physical processes and computer generations

Answer 3

only true randomness is generated by physical processess

Computer generated numbers can only give us pseudo random numbers out of a pregenerated list of many many numbers

Question 4

what is bootstrapping?

Answer 4

a statistical procedure that resamples a single dataset to create many simulate samples

this allows us to make population based estimates such as standard errors and confidence intervals without actually knowing the population data

this rests on the assumption that under large sample sizes the sample distribution will be approximatly normal and the standard deviation of the distribution will be equal to the standard error

more resamples derives a better estimate of the sampling distribution

Question 5

what is the probability of A given B

Answer 5

P(A|B) = P( A and B) / P(B)

Question 6

whats sensitivity and specificy in screening tests

Answer 6

sensitivity is likelyhood of finding a disease when present = P(Positive Test| Disease)
Specificity is likelyhood to give a negative result when disease is absent = P(Disease| Positive Test)

Question 7

what is bayes rule and how can we redefine it?

Answer 7

If we know P(A|B) but want to know P(B|A) we can use bayes rule
To use bayes rule we will also need to know the overall probability of B (known as the base rate)

Bayes rule:
P (B|A) = P(A|B) * P(B) / P(A)

Using the sum rule we can define P(A) as:
P(A) = P(A|B) * P(B) + P(A|¬B) * P(¬B)

therefore if we only have two outcomes bayes rule can also be expressed as:

P(B|A) = (P(A|B) * P(B)) / ( P(A|B) * P(B) + P(A|¬B) * P(¬B) )

Question 8

what are the names for the different parts of bayes rule

Answer 8

Bayes rule:
P (B|A) = P(A|B) * P(B) / P(A)

PRIOR= P(B)
LIKELYHOOD= P(A|B)
MARGINAL LIKELYHOOD= P(A)
POSTERIOR= P(B|A)

Question 9

How to write odds in probability notation?

Answer 9

odds of A = P(A) / P(¬A)

Question 10

how to write the odds ratio of bayes formula

Answer 10

Bayes rule:
P (B|A) = P(A|B) * P(B) / P(A)

Prior odds= P(B) / P(¬B)
Posterior odds = P (B|A)

odds ratio= Posterior odds/ Prior odds

an odds ratio of x shows us that the odds of B is increased x amount given A.

Question 11

what is the difference between the frequentist and the bayesian approach to statistics

Answer 11

FREQUENTIST APPROACH- interprets probabilities in terms of the long term frequencies after many repetitions. However this only works well for events that can be repeated many times in the same way

BAYESIAN APPROACH- reflects the degree of belief in a particular proposition

Question 12

what is a vector

Answer 12

an entity that has both direction and a magnitude but not a fixed position

Question 13

how would you notate vectors in space with coordinates

Answer 13

for 2D: p= xi + ij
for 3D:p= xi + yj + zk
where i j and k are vectors of length 1 along the x, y and z axes respectively

the vector can be visually represented as an arrow starting at an origin and ending at point p

Question 14

how do you add vectors geometrically?

Answer 14

• identify the parrelogram spanned by the two vectors

- the longest diagonal line between two parrelelogram points is the equivalent of vector1 + vector2

Question 15

what is a null vector?

Answer 15

the result of subtracting a vector from itself. this is the only vector that has zero magnitude

Question 16

what is a scalar

Answer 16

a physical quantity that is defined by its magnitude alone

Question 17

how to add vectors algebraicly

Answer 17

google it

Question 18

what does multiplying a vector by a scalar do

Answer 18

multiplying a vector by a scalar simply increases the length of the vector

google how to do it

Question 19

how to use pythagorases theorem to find the magnitude of a vector

Answer 19

m= square root of (a^2 + b^2)

Question 20

what are the inner products of vector multiplication

Answer 20

• a (dot) b
• which is equal to length of a | length of b | Cos(angle between a and b)
• this shows that the inner product of two vectors is a scalar
• it also shows that it is zero when the cosine of the angles between them is 0 (i.e when they are perpendicular
• therefore two vectors are perpendicular when their inner product is zero

Question 21

how does a magnititude of a vector relate to the inner product

Answer 21

the magnitude of a vector can also be expressed as the inner product of itself

Question 22

what is the difference between inner products and cross products of vector multiplication

Answer 22

• the inner product is defined in space whereas the cross product is only defined in 3D
• the inner product results in a scalar whereas the cross product results in a vector that is perpendicular to the plane spanned by the two original vectors
• a x b = c = a | b | sin(angle between a and b)