Lecture Three Flashcards
What is probability
A number between 1-0 that measures the likelihood that some event will occur.
what does 0 mean in terms of probability
more unlikely the event is bound to happen
what does 1 mean in terms of probability
it is very likely to happen
why is probability used
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
How to work out the probability
The outcome/all possible outcomes
What is a random experiment
process leading to two or more possible outcomes WITHOUT knowing exactly WHICH OUTCOME WILL OCCUR
what is the basic outcome
all the possible outcomes/realisations that could happen of a random experiment
e.g. a coin: heads or tails
a dice: 123456
Can wo basic outcomes occur simultaneously?
No
What must the random experiment do?
Lead to one of the basic outcomes
What is the sample space
A set of all basic outcomes
contains all the possible items including the items that cannot happen.
What is the symbol for a sample space
Ω
What is an event
a subset of basic outcomes from the sample space
When does an event occur?
If the random experiment results in one of its constituent basic outcomes
What is the nuff event
Represents the absence of a basic outcome
What is the nuff statement denoted by?
0
What is an independent event
each event is NOT affected by other events
(e.g. tossing a coin and getting head will always be the same probability 1/2)
What is a dependant event
An event affected by other events - it is conditional
What is an example of a mutually exclusive event
events cannot occur at the same time
e.g. heads and tails are mutually exclusive.
What is a random variable
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
an example of the random variable
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
What is a discrete random variable
a variable that takes on a countable number of values
What is a continuous random variable
one for which there are infinite possible outcomes between a range - the probabilities cannot be attached to specific outcomes
Describe how the continuous and discrete random variables will be displayed on a graph
Discrete will have lots of spaces between each data point - continuous wont
What does PDF stand for?
Probability Distribution Function
What is pdf
a mathematical function that describes the likelihood of a random variable taking on a given value
What type of probability distribution is used to find the pdf
discrete
What do we want to know when working with probability distribution for the discrete
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)
What is the representation of PDF in equation format
P(x) = P(X = x), ∀x
What does ∀ mean
all values of
What are the required properties of probability distribution
-Probabilities must NOT be negative or exceed 1
-Must always add up to equal 1
∑P(x) = 1
Why must all probabilities equal to 1
follows the fact that X= x, all possible values of x are MUTUALLY exclusive and Collectively exhaustive
What does CDF stand for?
Cumulative Probability Distribution
What is Cumulative Probability Distribution
a function that describes the probability distribution of a random variable
-the SUM of all probabilities that are smaller or equal to x
What is represented by CDF
F(x0)
How would u write CDF as a equation
F(x0) = P(X <= x)
Explain the equation of cdf
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
What is the relationship between PDF and CDF
Derived
Why are pdf and cdf a derived relationship
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
How can u get CDF from PDF
You can get the CDF from the PDF by adding up (integrating) all the probabilities from the start up to a certain value.
Where can u get PDF from the CDF
by finding the rate of change (differentiating) of the CDF. This tells you how the probabilities are distributed at each point
Where is the total probability on a graph
Total area UNDER the PDF curve is the Total probability and IS EQUAL TO 1
Can pdf or cdf be negative? Why?
No, both must be positive
-Pdf is the probability = must be positive
-Cdf ranges between 0-1
What does the word cumulative mean ?
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.
What are the derive properties of cdf
- cannot be less then 0, or bigger then 1
- 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
What are the axis for the cdf grapgh
- 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
What does the probability distribution contain
All information about the probability properties of a random variable. the graphical inspection of the distribution
What shape is the cdf graph
ladder/ steps
Why is the cdf shaped in this way
-they always increase
- e.g. when x Is 12 the cdf > when x is 11
What does E(X) stand for
expected value
What does expected value mean
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.
When happens to the expected value is the values are symmetrical
The expected value will be equal to the mean value
What is better expected value or mean
Expected value gives a much more of a precise answer
What is the purpose for the expected value
we need to compute estimate
What does the E do?
change 1/n (the usual mean) using P(x) to get the new weighted mean
What is the symbol for mean in regards to expected value
µ
What is the symbol for variance
σ2
What is the variance of a random variable
Expectation of the squared deviations about the mean
What is the symbol for standard deviation
σ
How to work out the standard deviation of a random variable
Square root of expected variance
How to computer expected variance
Exact same as normal except this time u multiply by the probability at the end
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
- Put in table on excel
- work out the mean: select all of px and x and write =SUMPRODUCT(all of x, all of px)
=2.15 - 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)
How to compute the skewness
(x-Mean)^3
How to computer kurtosis
(x-Mean)^4
What is a joint event?
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
What is used to compute the probability of a joint event
The Marginal Probability
What do businesses and economic applications of statistics concerned about?
The relationship between variables
Whats an example for which the marginal probabilities are used for?
Covariances
What does Joint probability distribution find
The joint probability finds the probability FOR BOTH VALUES
What are we interested in when looking at the joint probability distribution
THE INTERSECTION
What does MPD stand for
Marginal Probability Distribution
How is MPD obtained
Summing the joint probabilities over all possible values
equal to the probability of x / all values of y
What must all the MPD be equal to
1
P(x) = Sum of every
Properties of Joint distribution
-must be between 0-1
-equal to 1
What is Conditional probability distribution
-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!
What is the conditional probability distribution
P(y|x) = P(x,y) / P(x)
Explain Conditional Probability Distribution
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
How to calculate covariance of two random variables
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.
What does the covariance do
Find the relationship between 2 variables
What is a positive correlation and what does it mean to two random variables
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.
What is a negative correlation and what does it mean to two random variables
if Cov (X,Y) < 0
it implies that when X increases, Y tends to decrease. This suggests an inverse relationship.
What does 0 correlation and what does it mean to two random variables
If Cov(X,Y) = 0
It means there is no linear relationship between the variables; their variations are independent of each other.
How to get the covariance with excel
- calculate each mean for X and Y
- Calculate the (x- mean) and (y-mean) for each value
3.Multiply them both each result - =SumProduct(the inital values, then the values for no2)
What are the limitations to covariance for random variables
-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
What is better used in place of covariance
correlation
Why is that better used inplace of covariance
-provides a measure of strength of their liner relationship between two random variables
What is the correlation coefficient of two random variables equation
p = Cov(X,Y)
———–
σ xσ y
Annotate the formula
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.
Positive correlation of two random variables
(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
Negative correlation of two random variables
(−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
No correlation of two random variables
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.
What do linear sums and differences of random variables allow for
allow for flexible modelling of relationships between variables.
What are linwe sums and differences of random variables
refer to operations that combine two or more random variables using addition or subtraction.
What is the linear combination for random variables X and Y
Z=aX+bY
A and B are coefficents/constant
What is a coefficient/constant
a fixed value that is multiplied to a variable in a mathematical expression
What is the expected value of the linaer sums(difference) of two random variables
E[Z]=E[aX+bY]=aE[X]+bE[Y]
What does the expected value of the differences and linear sum equation mean
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
What is the variance of two random variables in the linear sum and differences
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
What is the variance if the covariance is = 0
Var(Z)=a^2 Var(X)+b ^2Var(Y)
