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Quantitive Methods > Lecture Three > Flashcards

Lecture Three Flashcards

1
Q

What is probability

A

A number between 1-0 that measures the likelihood that some event will occur.

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2
Q

what does 0 mean in terms of probability

A

more unlikely the event is bound to happen

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3
Q

what does 1 mean in terms of probability

A

it is very likely to happen

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4
Q

why is probability used

A

Because as humans and businesses, it is difficult to be certain about occurrence of future events - therefore businesses are able to use probability to make the best possible decisions

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5
Q

How to work out the probability

A

The outcome/all possible outcomes

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6
Q

What is a random experiment

A

process leading to two or more possible outcomes WITHOUT knowing exactly WHICH OUTCOME WILL OCCUR

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6
Q

what is the basic outcome

A

all the possible outcomes/realisations that could happen of a random experiment
e.g. a coin: heads or tails
a dice: 123456

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6
Q

Can wo basic outcomes occur simultaneously?

A

No

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6
Q

What must the random experiment do?

A

Lead to one of the basic outcomes

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6
Q

What is the sample space

A

A set of all basic outcomes

contains all the possible items including the items that cannot happen.

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6
Q

What is the symbol for a sample space

A

Ω

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6
Q

What is an event

A

a subset of basic outcomes from the sample space

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7
Q

When does an event occur?

A

If the random experiment results in one of its constituent basic outcomes

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7
Q

What is the nuff event

A

Represents the absence of a basic outcome

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7
Q

What is the nuff statement denoted by?

A

0

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7
Q

What is an independent event

A

each event is NOT affected by other events
(e.g. tossing a coin and getting head will always be the same probability 1/2)

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7
Q

What is a dependant event

A

An event affected by other events - it is conditional

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7
Q

What is an example of a mutually exclusive event

A

events cannot occur at the same time
e.g. heads and tails are mutually exclusive.

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7
Q

What is a random variable

A

a variable that takes on numerical values realized by the outcomes in the sample space generated by a random experiment

-a variable u can use but u dont know the value of the variable at the END of the period

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7
Q

an example of the random variable

A

return. u don’t know what return ur going to get at the end of the time period - u just make assumptions that it’ll be a positive return, but there’s no way of being certain

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7
Q

What is a discrete random variable

A

a variable that takes on a countable number of values

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7
Q

What is a continuous random variable

A

one for which there are infinite possible outcomes between a range - the probabilities cannot be attached to specific outcomes

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8
Q

Describe how the continuous and discrete random variables will be displayed on a graph

A

Discrete will have lots of spaces between each data point - continuous wont

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9
Q

What does PDF stand for?

A

Probability Distribution Function

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10
Q

What is pdf

A

a mathematical function that describes the likelihood of a random variable taking on a given value

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11
Q

What type of probability distribution is used to find the pdf

A

discrete

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12
Q

What do we want to know when working with probability distribution for the discrete

A

when working with discrete probability distribution we want to know the probability that the random variable (X) and the realisation of the random variable (x)

P(X = x)

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13
Q

What is the representation of PDF in equation format

A

P(x) = P(X = x), ∀x

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14
Q

What does ∀ mean

A

all values of

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15
Q

What are the required properties of probability distribution

A

-Probabilities must NOT be negative or exceed 1

-Must always add up to equal 1

∑P(x) = 1

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16
Q

Why must all probabilities equal to 1

A

follows the fact that X= x, all possible values of x are MUTUALLY exclusive and Collectively exhaustive

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17
Q

What does CDF stand for?

A

Cumulative Probability Distribution

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18
Q

What is Cumulative Probability Distribution

A

a function that describes the probability distribution of a random variable

-the SUM of all probabilities that are smaller or equal to x

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19
Q

What is represented by CDF

A

F(x0)

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20
Q

How would u write CDF as a equation

A

F(x0) = P(X <= x)

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21
Q

Explain the equation of cdf

A

F(x0) is the cdf and X is the random variable.
We want to find out the probability that the random variable isnt bigger or equal to a certain number which is x - u ADD them

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22
Q

What is the relationship between PDF and CDF

A

Derived

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23
Q

Why are pdf and cdf a derived relationship

A

Since X = x0 (which x0 is from CDF where F(x0) is the UNION of the MUTUALLY EXCLUSIVE X= x (from pdf) for all values of x less then or equal to x0.

The probability of this union is the SUM of these individual event probabilities

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24
Q

How can u get CDF from PDF

A

You can get the CDF from the PDF by adding up (integrating) all the probabilities from the start up to a certain value.

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25
Q

Where can u get PDF from the CDF

A

by finding the rate of change (differentiating) of the CDF. This tells you how the probabilities are distributed at each point

26
Q

Where is the total probability on a graph

A

Total area UNDER the PDF curve is the Total probability and IS EQUAL TO 1

27
Q

Can pdf or cdf be negative? Why?

A

No, both must be positive

-Pdf is the probability = must be positive
-Cdf ranges between 0-1

28
Q

What does the word cumulative mean ?

A

means increasing or growing by the addition of parts or elements

involves the idea of collecting or aggregating quantities over time or across a range.

29
Q

What are the derive properties of cdf

A
  1. cannot be less then 0, or bigger then 1
  2. the probability of a random variable does not exceed some number, cannot be more then the probability that doesnt exceed any larger number
    -cdf increases as x (pdf) increases
30
Q

What are the axis for the cdf grapgh

A
  • The horizontal axis (x-axis) represents the values of the random variable.
    The vertical axis (y-axis) represents the cumulative probability, which ranges from 0 to 1
31
Q

What does the probability distribution contain

A

All information about the probability properties of a random variable. the graphical inspection of the distribution

32
Q

What shape is the cdf graph

A

ladder/ steps

33
Q

Why is the cdf shaped in this way

A

-they always increase

  • e.g. when x Is 12 the cdf > when x is 11
34
Q

What does E(X) stand for

A

expected value

35
Q

What does expected value mean

A

using the mean by using the different weights

Expected value is the mean, but the mean isnt equal for each value- it is weighted depending on the value.

36
Q

When happens to the expected value is the values are symmetrical

A

The expected value will be equal to the mean value

37
Q

What is better expected value or mean

A

Expected value gives a much more of a precise answer

37
Q

What is the purpose for the expected value

A

we need to compute estimate

38
Q

What does the E do?

A

change 1/n (the usual mean) using P(x) to get the new weighted mean

39
Q

What is the symbol for mean in regards to expected value

A

µ

40
Q

What is the symbol for variance

A

σ2

41
Q

What is the variance of a random variable

A

Expectation of the squared deviations about the mean

42
Q

What is the symbol for standard deviation

A

σ

43
Q

How to work out the standard deviation of a random variable

A

Square root of expected variance

44
Q

How to computer expected variance

A

Exact same as normal except this time u multiply by the probability at the end

45
Q

Explain in steps what to do to computer expected variance on excel using values given x = 0,1,2,3,4,5
P(x) = 0.3, 0.2 ,0.1 ,0.05 ,0.15 ,0.2

A
  1. Put in table on excel
  2. work out the mean: select all of px and x and write =SUMPRODUCT(all of x, all of px)
    =2.15
  3. Then get each value from x and Minus the mean and square it by 2
    =(0 - 2.15)^2 = 4.6225
    -do for every value of x
    -Then =SUMPRODUCT(values of px, sum of all the results from point 3)
46
Q

How to compute the skewness

A

(x-Mean)^3

47
Q

How to computer kurtosis

A

(x-Mean)^4

48
Q

What is a joint event?

A

the value A moves at the same time as the value B

-they move together & have the joint probabilities, they move in the same direction

49
Q

What is used to compute the probability of a joint event

A

The Marginal Probability

50
Q

What do businesses and economic applications of statistics concerned about?

A

The relationship between variables

51
Q

Whats an example for which the marginal probabilities are used for?

A

Covariances

52
Q

What does Joint probability distribution find

A

The joint probability finds the probability FOR BOTH VALUES

53
Q

What are we interested in when looking at the joint probability distribution

A

THE INTERSECTION

54
Q

What does MPD stand for

A

Marginal Probability Distribution

55
Q

How is MPD obtained

A

Summing the joint probabilities over all possible values

equal to the probability of x / all values of y

56
Q

What must all the MPD be equal to

A

1

57
Q
A

P(x) = Sum of every

58
Q

Properties of Joint distribution

A

-must be between 0-1
-equal to 1

59
Q

What is Conditional probability distribution

A

-probability of x being realised but on the condition of something else happening

  • allows you to understand how the likelihood of one event changes when you know that another event has occurred. It helps us make more informed predictions based on available information!
60
Q

What is the conditional probability distribution

A

P(y|x) = P(x,y) / P(x)

61
Q

Explain Conditional Probability Distribution

A

The conditional probability of random variable Y given that random variable X takes the value x, expresses the probability that Y takes the value y as a function of y, when the value x is fixed for X

X given Y = y

62
Q

How to calculate covariance of two random variables

A

Cov(X,Y)=E[(X−E[X])(Y−E[Y])]

:

E[X] is the expected value (mean) of 𝑋

E[Y] is the expected value (mean) of

E denotes the expectation operator.

63
Q

What does the covariance do

A

Find the relationship between 2 variables

64
Q

What is a positive correlation and what does it mean to two random variables

A

if Cov (X,Y) > 0

it indicates that when X increases, Y tends to increase as well.
This suggests a direct relationship between the two variables.

65
Q

What is a negative correlation and what does it mean to two random variables

A

if Cov (X,Y) < 0

it implies that when X increases, Y tends to decrease. This suggests an inverse relationship.

66
Q

What does 0 correlation and what does it mean to two random variables

A

If Cov(X,Y) = 0

It means there is no linear relationship between the variables; their variations are independent of each other.

67
Q

How to get the covariance with excel

A
  1. calculate each mean for X and Y
  2. Calculate the (x- mean) and (y-mean) for each value
    3.Multiply them both each result
  3. =SumProduct(the inital values, then the values for no2)
68
Q

What are the limitations to covariance for random variables

A

-it doesnt have an upper/lower bound -size is influenced by the scalling of the numbers
-doesn’t provide a standardized measure of strength or direction

69
Q

What is better used in place of covariance

A

correlation

70
Q

Why is that better used inplace of covariance

A

-provides a measure of strength of their liner relationship between two random variables

71
Q

What is the correlation coefficient of two random variables equation

A

p = Cov(X,Y)
​ ———–
σ xσ y

72
Q

Annotate the formula

A

Cov(X,Y) is the covariance between variables X and Y.

𝜎X is the standard deviation of 𝑋.

σ Y is the standard deviation of Y.

73
Q

Positive correlation of two random variables

A

(0<p≤1)
-Indicates a direct relationship; as one variable increases, the other tends to increase as well.

A value of 1 indicates a perfect positive linear relationship

74
Q

Negative correlation of two random variables

A

(−1≤p<0): Indicates an inverse relationship; as one variable increases, the other tends to decrease.

A value of -1 indicates a perfect negative linear relationship

75
Q

No correlation of two random variables

A

p=0): Suggests that there is no linear relationship between the two variables.

However, it’s important to note that this does not imply that the variables are independent; they could have a non-linear relationship.

76
Q

What do linear sums and differences of random variables allow for

A

allow for flexible modelling of relationships between variables.

77
Q

What are linwe sums and differences of random variables

A

refer to operations that combine two or more random variables using addition or subtraction.

78
Q

What is the linear combination for random variables X and Y

A

Z=aX+bY

A and B are coefficents/constant

79
Q

What is a coefficient/constant

A

a fixed value that is multiplied to a variable in a mathematical expression

80
Q

What is the expected value of the linaer sums(difference) of two random variables

A

E[Z]=E[aX+bY]=aE[X]+bE[Y]

81
Q

What does the expected value of the differences and linear sum equation mean

A

This property shows that the expected value of the sum (or difference) of random variables is equal to the sum (or difference) of their expected values, scaled by the respective constants

82
Q

What is the variance of two random variables in the linear sum and differences

A

variance z is given by:

Var(Z)=Var(aX+bY)=a^2Var(X)+b ^2
Var(Y)+2abCov(X,Y)

Where:

Var(X) and Var(Y) are the variances of X and Y.

Cov(X,Y) is the covariance between X and
Y, which measures how the two random variables vary together

83
Q

What is the variance if the covariance is = 0

A

Var(Z)=a^2 Var(X)+b ^2Var(Y)

Quantitive Methods (6 decks)

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