Probability Flashcards
What is a random experiment?
a process that results in a number of possible outcomes, none of which can be predicted with certainty;
What is the sample space?
sample space of a random experiment is a list of all possible outcomes
E.g. roll a die: sample space:
S={1, 2, 3, 4, 5, 6}.
How can variance and standard deviation be used in real life?
Higher standard deviation/variance = higher return
What is a mutually exclusive outcome?
When two events cannot occur at the same time in a single trial
Therefore P(A) x P(B) does not equal to P(AxB)
What is a collectively exhaustive outcome?
one of the events must occur AKA one or the other
eg. heads and tails in a coin toss is collectively exhaustive because one of them must occur. If head does not occur, tails must occur
What is the probability of sample space?
P(S) =1
What is P(A|B)?
Conditional probability that A occurs, given that B has occurred:
What is P(A or B) = P(A U B) = P(A union with B)?
A occurs, or B occurs, or both occur
P(A or B) = P(A) or P(B) - P(A and B)
What is P(A and B) = P(A ∩ B) = P(A intersection with B) ?
A and B both occur
What is another way to express P(A)?
P(A) = P(A∩B) + P(A ∩Bc)
What is another way to express P(A or B)?
P(A or B)=P(A)+P(B)-P(A and B)
How to test if two events are independent?
P(A|B)=P(A) or P(B|A)=P(B)
P(A and B) = P(A)*P(B)
Prove that P(A|B) = P(A)
= [P(A and B)]/P(B)
=[P(A)*P(B)] /P(B)
=P(A)
what is marginal probability
Computed by adding across rows or down columns - calculated in the margins of the table
Prove the complement rule
Given an event A and its complement, Ā, so that A+ Ā=S;
Know that P(S)=1;
so P(A)+P(Ā)=1;
therefore P(Ā)=1-P(A)
Another way to express P(A|B)
another way to express P(B|A)
P(A|B) = P(A and B)/ P(B)
P(B|A) = P(B and A)/ P(A)
another way to express P(A and B)
prove
as P(A|B) = P(A and B) / P(B)
P(A and B) = P(A|B) x P(B)
formula for expected value
random variable example
Imagine tossing three unbiased coins.
S= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT)
8 equally likely outcomes.
Let X = number of heads that occur.
X can take values 0, 1, 2, 3.
how to denote random variable and actual realised values?
Denote random variables (X, Y, etc) in upper case
Denote actual realised values (x, y etc) in lower case
Example: X is the random variable that can take values 0, 1, 2, 3.
Actually perform experiment, find the pattern HTT. Then x=1.
what is a discrete random variable?
discrete random variable has a countable number of possible values, e.g. number of heads, number of sales etc.
Does not have to be finite; but values can be strictly ordered.
based on counting process
what is a continuous variable?
continuous random variable has an infinite number of possible values
based on measuring process
what are features of probability distribution?
Sum of proabilities must equal 1
P(X=x)
what are frequentist probabilities?
Probabilities associated with r.v.s are often associated with relative frequencies.
E.g. interested in number of photocopiers sold by different firms. Estimate probabilities from frequencies
what is the probability disturibution for a discrete variable?
mutually exclusive list of all possible numerical outcomes along with probability of occurence of each outcome
what are numerical variables?
how can they be further categorised?
numerical variables produce numerical repsonses
further categorised into discrete and continuous numerical variables
What are rules for expectations? given that c is a constant
E(c)=c
E(cX)=cE(X)
E(X-Y)=E(X)-E(Y)
E(X+Y)=E(X)+E(Y)
E(XY)=E(X)*E(Y) only if X and Y are independent
2.Let Z=3X+2Y-2XY+3, with E(X)=3, E(Y)=5, X and Y independent
calculate Z
= E(3X+2Y-2XY+3)
=3E(X)+2E(Y)-2E(X)*E(Y)+3
=3*3+2*5-2*3*5+3
=9+10-30+3
= -8
what does variance measure?
spread/dispersion of distribution
what are laws for variances?
- V(c)=0
- V(cX)=c²V(X)
- V(X+c)=V(X)
- V(X+Y)=V(X)+V(Y) if X and Y are independent
- V(X-Y)=V(X)-V(Y) if X and Y are independent
a) Find the mean and variance of X
b) Find the mean and variance of Y = 2*X+5
what is bivariate distribution?
distnguish with univariate distribution
Distribution of a single variable – univariate
Distribution of two variables together – bivariate
So, if X and Y are discrete random variables, then we say p(x,y) = P(X=x and Y=y) is the joint probability that X=x and Y=y.
show a bivariate distribution table
Let X be the number of heads.
Let Y be the number of changes of sequence, i.e. the number of times we change from H →T or T→H
how can you tell if random variables are independent?
If the random variables X and Y are independent, then
P(X=x and Y=y) = P(X=x) * P(Y=y)
p(x,y) = px(x) * py(y)
how do you calculate sum of two random variables?
by using bivariate distribution
Then, P(X+Y=2) = sum of all joint probabilities for which x+y=2;
Continue to do for P(X+Y=3), P(X+Y=4) Show
P(X+Y=2)= p(0,2) + p(1,1) + p(2,0)
= 0.07 + 0.06 + 0.06
=0.19
…continue for P(X+Y=3), P(X+Y=4)
evaluate mean and variance of (X+Y)
E(X+Y) = 1.2
V(X+Y) = 0.56
what happens when joint distribution of X and Y is unknown?
use the rules learnt earlier:
If a and b are constants, then
E(aX+bY) = aE(X) + bE(Y)
V(aX+bY) = a²V(X) + b²V(Y) – only if X and Y are independent!
E(X) = µx and E(Y)= µy, then the covariance between X and Y is given by
what is the formula for variance when there is correlation?
how does correlation relate to investment risk?
Analysts reduce risk by diversifying their investments – that is, combining investments where the correlation is small.
An investor forms a portfolio by putting 25% of his money in Stock A and 75% in stock B, with parameters below.
what is the expected return of the portfolio?
E(RA)=0.08, E(RB)=0.15
Rp=0.25*RA+0.75*RB
E(Rp) = 0.25*E(RA)+0.75E(RB)
= 0.25*0.08+0.75*0.15
= 0.1325
Expected return of the portfolio is 13.25%.
Determine the variance of the return if ρ=1
Counting rule 2: what do you do when you have there are k1 events on the first trial, k2 events on the second trial…and kn events on the nth trial then the possible outcomes
(k1)(k2)…(kn)
Counting rule 3: how do arrange number of n items in order?
N!
Counting rule 4: when an entire subject of an entire group of items needs to be arranged in order
Permutation : n!/ (n-x)!
Counting rule 5: select subset from entire group of items without the need for order
Combination: n! (n-x)! x!
What is a Bernoulli process?
Two possible outcomes for each trial, labelled success and failure
Probability of success is p, probability of failure is (1-p)
Trials are independent – the outcome of one trial does not affect the outcomes of any other trials.
What properties does a binomial experiment have?
- A fixed number of trials, n
- Two possible outcomes for each trial, labelled success and failure
- Probability of success is p, probability of failure i
- Trials are independent – the outcome of one trial does not affect the outcomes of any other trials.
What is the binomial random variable?
binomial random variable is defined as the number of “successes” in the n trials
How to express the probability that we will have k successes?
X~Bin(n=….,p…..)
Eg. n =12 p =3
Then X~Bin(n=12,p=0.25)
Show mean and variance of binomial distribution
guideline for using binomial table
Note that P(X ≤0)= P(X=0)
P(X ≤k) →use tables
P(X=k) = P(X≤k) – P(X≤(k-1))
P(X
P(X≥k) = 1- P(X≤[k-1])
P(X>k) = 1- P(X ≤k)
features of bivariate distributions
Marginal distributions
Independence
Sum of two random variables
Covariance and Correlation
Linear Combinations of Variables
features of binomial distribution
binomial experiment
bernoulli trial
binomial formula
mean and variance of a binomial variable
binomial tables.
what does expected value show?
long term average of discrete random variable
formula for variance
E(X2) - [E(X)]2
what are marginal distribution functions?
In the casino game roulette, a gambler can bet on which of
38 numbers will be the result when the roulette wheel is
spun. On a $2 bet, a gambler gains $70 if he or she picks the
right number, but loses the $2 otherwise.
a. Let X = amount gained or lost on a $2 bet on a roulette
number. Write out the probability distribution of X.
In the casino game roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number, but loses the $2 otherwise
Imagine a player decides to bet on the same number for 10 trials in a row, and count Y = the number of wins in those 10 trials. What sort of distribution does Y have? What are the mean and variance of this
distribution?
Hence, Y will be a binomial variable.
Y~Bin (n=10, p=1/38)
E(Y) = np = 10/38 Var(Y) = np(1-p) =10\*1/38\*37/38=370/1444= 185/722
In the casino game roulette, a gambler can bet on which of 38 numbers will be the result when the roulette wheel is spun. On a $2 bet, a gambler gains $70 if he or she picks the right number, but loses the $2 otherwise
. Again considering Y, the number of wins in 10 trials,
what is the probability that the player wins on at least
5 trials? What is the probability that there are no wins
in the 10 trials?
P(Y≥5) = 1-P(Y≤4) = 1- 1.0000 = 0 P(Y=0) = 0.765916
10C0 x (1/38)0x (37/38)10
