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Quantitive Methods > Lecture Five > Flashcards

Lecture Five Flashcards

1
Q

What is discrete probability

A

When there are a finite number of results

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2
Q

An example of a discrete probability

A

Flipping a coin: 2 outcomes heads or tails

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3
Q

What is an example of continuous probability

A

Normal probability

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4
Q

What can be calculated once a probability distribution is constructed

A

a subset of the real number space

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5
Q

Why is it difficult to identify individual outcomes in the real world

A

Bc the sample is complex and large

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6
Q

Why are there many types of probability distributions and the result of this

A

for various types of random variables as analytic functions

=we are able to calculate probabilities for events of interest without construction the probability distribution ourselves

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7
Q

What is the pdf for a random variable

A

Probability distribution function describes how probabilities are distributed over the values of random variables

-tells us the probability of a particular value coming out

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8
Q

What is pdf also known as

A

Probability mass function

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9
Q

What is the cdf for a random variable

A

The Cumulative distribution function of a random variable is a way of describing how probabilities are distributed over the values of a random variable

-tells us how likley the value is to be less then some other value

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10
Q

What would cdf be

A

a sum over a range
-sum of pdf = cdf

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11
Q

What is the Bernoulli model a building block for

A

binomial distribution

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12
Q

What does the Bernoulli model answer

A

The question if it is a success or a fail

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13
Q

What type of data does the Bernoulli model take

A

discrete
-takes one of two outcomes to determine if its a fail or success - no inbetween

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14
Q

What must the sum of the Bernoulli model be

A

The two probabilities must be equal to 1

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15
Q

What would the value of success be and the value of failure

A

Success = 1
P(1) = p

Failure = 0
P(0) = 1-p

-want to find out what P is meaning P(X) - we want to find x

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16
Q

How to compute the mean of the Bernoulli model

A

E(X) = p

x(when failure,0) = 1-p + 1(p) = p

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17
Q

How to compute the variance for the Bernoulli model

A

Probability of success * probability of failure

p*1-p

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18
Q

What is the Bernoulli model based on

A

One specific trial

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19
Q

How to work out the mean and variance if the probability of success is 0.4

A

Mean is 0.4
-because mean = p

The variance: 1-0.4 = 0.6
0.6 x 0.4 = 0.24
v = 0.24
-because variance = p*(p-1)

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20
Q

What is the binomial distribution?

A

important generalization of the Bernoulli distribution

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21
Q

What are the properties of the binominal distribution?

A

1- the experiment consists of a sequence of n identical trials (more then 1 trial/round)

2- it has 2 outcomes: success/failure for each trial

3- The probability of success is denotated by p. it doesnt change from each trial

4- The trials are independent of each other

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22
Q

What are we interested in for the binomial distribution

A

The number of successes occurring in the n trials

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23
Q

What is denoted for the number of sucesses

A

X

24
Q

Formula of binominal distribution

A

=nCx x px (1-p)n-x

n: The number of experiments

p: The probability of success in a single experiment

q: The probability of failure in a single experiment, which is equal to 1 - p

nCx: The combination of n and x

25
Q

Mean of the binominal distribution

A

Probability * number of trials

26
Q

Variance of binomial distribution

A

p*p-1 adjusted to number of trials

27
Q

What is the formula of the binominal distribution using combinations

A

(n!/(n-x)!x!)) * p^x(1-p)^n-x

(n!/(n-x)!x!)) : refers to the number of experimental outcomes providing exactly x successes in n trials

p^x(1-p)^n-x : is the probability of a particular sequence of trial outcomes with x successes in n trials

28
Q

What does the binominal distribution formula for combinations do?

A

Will get u all the ways u can get a success
-all the combos to get successes are computed here
-then this probability is adjusted for the number of trials and number of successes

29
Q

How to compute the binominal formula on excel and explain each element

A

=BINO.DIST(Number_S, Trials, Probability_s, Cumlative)

-Number_s = the number of success in trials the x in formula)

-Trials = the number of independent trials (n in the formula)

-Probability_s = the probability of success

-Cumulative is the logical value that determines the form of the function:
-true = Bino.dist returns the cumulative distribution function(cdf)
-false = Bino.dist returns the probability distribution formula (pdf)

30
Q

How to work out the mean of the Binomial Probability Distribution

A

E(X) = np

=the probability x number of trials

-the mean of Bernoulli (which is also equal to p) times the trials

31
Q

How to work out the variance of the Binomial Probability Distribution

A

mean * 1-p

=np*(1-p)

32
Q

How to work out the standard deviation of the binominal distribution

A

square root of np*(1-p)

-square root of the variance

33
Q

How does the variance link to the success

A

smaller variance = closer to success
larger variance = closer to failure

34
Q

When do u use the binomial distribution

A

-when there are two outcomes: yes/no, true/false etc

35
Q

What to check before computing the binomial distribution

A

1- application has many trials that only have to outcomes

2- probability is the same for each trial

3- the probability of one trial doesn’t affect the probability of other trials

36
Q

What do u do if u want to find less then a certain value of probabilities for the binominal distribution on excel

A

want to work out a e.g. P(less then 2)
-then that means P(x<2) ALSO means P(x≤1)
-because binomial is discrete: cant take values that are decimal
so:
=BINOM.DIST(1,n,p,TRUE)

-1 because u want to find stuff less then or equal to 1
-want 0,1

37
Q

What do u do if u want to find more then a certain value of probabilities for the binominal distribution on excel

A

-want to work out e.g. p(more then 4)
-then means P(x>4) MEANS 1-P(x≤4)
-takes away the smaller numbers before 4 and
So:
=1-BINOM.DIST(4,n,p,TRUE)

-minus bc u dont want the values before 4. only above it
-want 4,5,6,etc

38
Q

What do u do if u want to find at least certain value of probabilities for the binominal distribution on excel

A

-want to work out P(at least 3)
-want nothing less then 3
-P(x ≥ 3) ALSO MEANS 1-P(x≤2)
So:
=1-BINOM.DIST(2,n,p,TRUE)

because want to work out nothing less then 3, want 3,4,5 etc

39
Q

what random variable numbers are in continuous distribution

A

take any possible specific value inside a range

40
Q

are there limited numbers in continuous

A

can create infinity numbers, e.g. between 0-1 u can have 0.1,0.2,0.3 but also 0.11,0.12 etc

41
Q

What is important when working out the continuous probability distribution

A

compute probability using continuous random variables - not caring about the specific values, only the range of values
if random variable x is between 2 numbers
-and the probability will be that the x is between these two numbers

42
Q

what does CDF and probability express

A

expresses that X does not exceed the value of x :

F(x) = P(X bigger/equal to x)
-it takes values between ranges

Cumulative x takes a range of values contained in this

43
Q

How to find the probability that continuous random variables fall into a specific range

A

-need to find difference between the CDF at the upper end of the range and the CDF at the lower end of the range
P(a< X < b)
=
F(b)- F(a)

44
Q

Why do we subtract upper from lower in trying to find the probability range using cdf

A

because cumulative is always increasing L->R
-if b>a then the cumulative function F(b) has to be > then the cumulative function of F(a)
=because b contains the probability of a that was in the past
-has to be at least that value OR larger

45
Q

Explain in graph how we would find the specific range

A

the upper side - the end side = the middle area

46
Q

probability of random variable assuming a value is within some range

A

=pdf (probability density function) the curve under

47
Q

What are the two common continuous distribution

A

Uniform and Normal

48
Q

What is Uniform distribution

A

it will give the same probability to every single observation inside the range

49
Q

when is uniform distribution used

A

When probabilities of outcomes in the same sample space are the same

50
Q

What is the pdf of uniform distribution

A

1/b-a

upper bound - lower bound / 1

51
Q

what is the mean of pdf uniform distribution

A

a+b / 2

-difference / 2

52
Q

What is the variance of the pdf uniform

A

squared difference f the upper and lower bound/12

(b-a)^2 / 12

53
Q

What to do if want to find a new range of values for the pdf and uniform distribution

A

new range has to be inside of the old range

d-c/b-a

54
Q

What is normal distribution

A

most important
-describes continuous disitribtion

55
Q

What is normal distribution used for

A

Approximating bionominal distribution

-everything is used by modelling normal distribution e.g dna test/weight

56
Q

What shape is the graph

A

bell

57
Q

What are the mean reasons for normal distribution wide application

A

1- closely approximates the probability distributions of a WIDE RANGE of random variables

2- Distributions of sample means approach a normal distirbution GIVEN LARGE sample size

3- computation of probabilities are direct and elegant

4- Lead to good business decisions due to the number of applications

Quantitive Methods (6 decks)

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