Please enable JavaScript.
Coggle requires JavaScript to display documents.
Chapter 7 - Random variables and discrete probability distributions -…
Chapter 7 - Random variables and discrete probability distributions
Random variable
The value of a random variable is a numerical event
1
A function that assigns a number to each outcome of an experiment
Probability distributions
We can describe them by computing various parameters
4
Discrete probability distribution represents a population
5
Discrete bivariate distributions
11
12
The requirements for a bivariate distribution are:
Independence
COV (X;Y) = 0
17
Discrete random variable
Takes on a countable number
e.g. integers
Probability distributions
A table, formula, or graph that describes the values of a random variable and the probability associated with these values
We have two types of probability distributions:
Discrete probability distributions
Continuous probability distributions
Bivariate distributions
Also called joint probabilities
A joint probability distribution of X and Y is a table of the values of x and y and is denoted
P(x;y)
Probabilities of combinations of two variables
Rules of expected values
Rule 1
E(c) = c
The expected value of a constant c is just the value of the constant
Rule 2
E(X+c) = E(X) + c
Rule 3
E(cX) = cE(X)
We can pull a constant out of the expected value expression
Population variance
Weighted average of the squared deviations from the mean
8
Calculation formula
9
Standard deviation
10
Notation
Probability of the random variable X will equal x is:
P(X=x)
or simply P(x)
The lowercase counterpart, x, will represent the VALUE of the random variable
Uppercase letter will represent the NAME of the random variable usually X
Rules of variance
Rule 1
V(c) = 0
The variance of a constant c is zero
Rule 2
V(X+c) = V(X)
The variance of a random variable and a constant is just the variance of the random variable
Rule 3
V(cX) = c^2 V(X)
The variance of a random variable and a constant coefficient is the coefficient squared multiplied with the variance of the random variable
Continuous random variable
Values are not discrete, not countable
e.g. real numbers
Marginal probabilities
We can calculate the marginal probabilities by summing across rows and down columns to determine the probabilities of X and Y individually
Covariance
The calculation formula
14
Which is equivalent to:
15
13
Measures the strength of the linear relationship between two variables
Hypergeometric distributions
n = sample size
X = number of successes in the sample
N - A = number of failures in the population
n - X = number of failures in the sample
A = number of success in the population
The mean of the hypergeometric distribution is:
N = population size
24
23
The standard deviation is
Outcomes of trials are dependent
25
Sample taken without replacement
Where 26 is called the finite population correction factor from sampling without replacement from a finite population
n trials in a sample taken from a finite population of size N
Probability distributions
18
Coefficient of correlation
16
Application in finance
Portfolio expected returns and portfolio risks
Binomial distributions
Two sampling methods
The number of combinations of selecting X objects out of n objects is:
Observations are independent
19
Constant probability for each observation
20
Two mutually exclusive and collectively exhaustive categories
21
A fixed number of trials (observations), n
22
Bivariate distributions
We describe the mean, variance and standard deviation of each variable in a bivariate distribution by working with the marginal probabilities
Rules for the sum of two variables
Rule 2
V(X+Y) = V(X) + V(Y) + 2COV(X;Y)
Rule 3
If X and Y are independent
V(X+Y) = C(X) + X(Y)
Rule 1
E(X+Y) = E(X) + E(Y)
Sum of two variables
The bivariate distribution allows us to develop the probability distribution of any combination of the two variables, of particular interest is the sum of the two variables
Population mean (expected value)
Weighted average of all its values
The weights are the probabilities
6
7
Discrete probability distributions
2
3
The probabilities of the values of a discrete random variable may be derived by means of any applicable technique, the following two conditions must apply: