Joint Distribution, Measure of Error & Families of Discreet Random Variables

Question 1

E(x+b) =

Answer 1

E(x) + b

Question 2

E(a*x) =

Answer 2

E(x) * a

Question 3

E(a*x+b) =

Answer 3

a*E(x) + b

Question 4

var(x+b) =

Answer 4

var(x)

Question 5

var(a*x) =

Answer 5

(a^2)* var(x)

Question 6

var(a*x+b) =

Answer 6

(a^2)*var(x)
- no b

Question 7

What is covariance?

Answer 7

Covariance measure how much two random variables change together.
Covariance only describes linear relationships between two variables.

Question 8

Cov(x,y) =

Answer 8

E(x) - E(x)*E(y)

Question 9

correlation coefficient

Answer 9

cov(x,y) / σ(x)*σ(y)

Question 10

E(x+y) =

Answer 10

E(x) + E(y)

Question 11

A higher dimension of binomial distribution
- fixed # of trials (n)
- all trials (xi) are independent
- for each trial (xi) there are k categories (x1, x2, …. xk)
- probabilities of occurring are p1, p2, … pk where p1 + p2 + … pk = 1

Answer 11

multinomial distribution

p(X = x1) = [ n! / ( x1! * x2! * …xk! ) ] * p1^x1 * p2^x2 * … pk^xk

Question 12

bias =

Answer 12

(mean) - (true value)
Bias is the systematic error, which is the same for every measurement and may be the result of a lack of calibration.

Question 13

accuracy

Answer 13

Accuracy is the closeness of the measurements to the true value, determined by the bias.

Question 14

computation of uncertainty

Answer 14

var(some formula with constants and variables)
(all constants part)^2 * var(variable)
(all constants part)^2 *(σ)^2
σ = sqrt((all constants part)^2 *(σ)^2) = (all constants part) *(σ)
NOTE: take partial derivatives when more than one variable is involved (such as height and radius), ADD those terms, do the square root thing to find the uncertainty

Question 15

What is propagation of error?

Answer 15

finding the uncertainty of measurements

Question 16

What is propagation of error?

Answer 16

finding the uncertainty of measurements

Question 17

All possible values have the same probability

Answer 17

Discrete Uniform: parameters are (a, b)
E(x) = (a + b) / 2

var(x) = [ (b - a)*(b - a + 2) ] / 12

but also var(x) = E(x^2) - [E(x)]^2

Question 18

Any experiment with only two outcomes

Answer 18

Bernoulli Trials
A Bernoulli random variable is described as x = 1 if ‘success’, or x = 0 if ‘failure’
p(x) = p if success
p(x) = 1-p if failure

Question 19

A series of Bernoulli random variables

Answer 19

binomial random variable
n = # of trials
p = probability of success

Question 20

A large number of trials with a small probability of occurring

Answer 20

Poisson random variable
x = # of occurrences over a specified interval or region
EXAMPLES: calls in an hour, # of chocolate chips in your cookie
λ: avg # of occurrences per unit time
t = # of time units over a particular interval
NOTATION: x ~poisson(λt)

Question 21

var(x + y) =

Answer 21

var(x) + var(y) =

Question 22

Monitor over a time interval
RV is x ~ ____________(λ,t)

Answer 22

Poisson random variable
x: the number of occurrences over a time interval