chapter 2

Question 1

random variable

Answer 1

formal statistical model for representing the results of an experiment

simply numeric representation of the result on an experiment

its a function x that that associates a unique numerical value with every outcome in the sample space

not lunique, for a given experiment there are infinitely many

each time an experiment is conducted an o observation of the random variables are made

traditional to denote random variable as capital letter and its realisation as lower case equivalent

Question 2

discrete random variable

Answer 2

is a random variable whose sample space only has a countable (finite or countably infinite)) number of outcomes

Question 3

sample space is countable if

Answer 3

outcomes can be written in size order

Question 4

probability mass function

Answer 4

method of specifying the probability of each outcome(
pmf) p(x)

specifies the probability that the random variable X=x(si) for all outcomes r(si) in the sample space
used for discrete random variables

must satisfy:
p(x)>o for al outcomes x - cannot have -ve probability
sum of probabilities for all outcomes must equal 1

Question 5

expectation

Answer 5

E[X]=x1 x p1 + x2 x p2 +…+xn x pn

Question 6

variance

Answer 6

measure of how spread out a set of data is

Question 7

cumulative distribution function

Answer 7

represents the probability of a realisation from a random variable is less than or equal to x

f(X)=P(X is less than or equal to x)

is defined for all real numbers not just element of sample space
the CDF for any other outcome x can be calculated by adding up the probabilities that are less than or equal to X

Question 8

probability mass function

Answer 8

A probability mass function (PMF) p(x) specifies the probability that the random
variable X = x(si) for all outcomes x(si) in the sample space. p(x)=P(X =x(si))=P(X =x).
must satisfy two fundamental properties:
probabilities must be greater than 0 and the sum of all probabilities must equal 1

Question 9

E(X+Y)=

Answer 9

E(X) + E(Y)

Question 10

E(a)=

Answer 10

a

Question 11

E(aX) =

Answer 11

aE(X).

Question 12

E(aX+b)=

Answer 12

aE(X)+b.

Question 13

if independant E(XY)=

Answer 13

E(X)E(Y)

Question 14

standard deviation

Answer 14

square root of the variance

Question 15

var(a)=

Answer 15

0

Question 16

var(X) =

Answer 16

greater than or eequal to 0

Question 17

var(X)=0 only if…

Answer 17

Xis a constant

Question 18

var(bX)=

Answer 18

b2Var(X)

Question 19

var(bX+a)=

Answer 19

b2Var(X)

Question 20

variance: if X and Y are independent in variance

Answer 20

Var(X+Y)=Var)X) + Var(Y)

Var(aX + bY ) = a2Var(X) + b2Var(Y )

Question 21

var(X-Y)=

Answer 21

= Var(X) + Var(Y ).

Question 22

discrete distributions

Answer 22

probability models for discrete random variables

Question 23

Descrete distributions: Bernoulli

Answer 23

simplest descrete districutions
models situations where there are only two possible outcomes referred to as 0 and one and fail or successs respectively
probability of a success is equal to 0<1
denoted by X tilda Bern(thita)

Question 24

Bern: P(success)=

Answer 24

thita

Question 25

Bern: P(fail)

Answer 25

1-thita

Question 26

Bern: pmf can be written as

Answer 26

p(x) = P(X = x) = thita to power of x x(1- thita)to power of1-x here that is a perimeter which is likely unknown but must be between 0 and 1 would estimate the value of that from data

Question 27

Bern: expectation and variance

Answer 27

exp turns out t be that | var turns out to be that(1- that)

Question 28

Binomail distribution

Answer 28

represents the probability of successes in a number of identical and independent Bernoulli trials denoted by X tilda Bin(number of trials, thita) so X= the sum of the n Bernoulli trials X represents the number of successes

Question 29

Bin: pmf

Answer 29

p(x)=P(X=x)= nCx thita to power x (1- thita) to power n-x

Question 30

Bin: coefficients nCx

Answer 30

nCx= n!/(n-x)!x!

Question 31

Bin: three secial cases

Answer 31

p(X=0) = (1- thita) to power n... no successes p(X=n) = thita to power n...successes only p(X greater than or equal to 1) = 1-(1- thita) to power n... at least one success

Question 32

Bin: expectation an variance

Answer 32

``` E(X)= n x thita Var(X)= nthita(1- thita) ```

Question 33

Geometric districution

Answer 33

models the number of Bernoulli trials required to obtain the first success denoted by: X tilda Geo (thita)

Question 34

geometric: pmf

Answer 34

p(x)=P(X=x)=(1- that) to power ofx-1 all times thita forx=1,2,3,... has infinitely many outcomes instead of just 0 and 1 because its thenumber of failures it takes to get a success sum of all probabilities still equals one though

Question 35

geometric: mean and variance

Answer 35

...

Question 36

poisson distibution

Answer 36

denotes the number of events that occur randomly in a fixed point in time or space