12 Discrete Random Variables Flashcards
Random variable
A variable that takes on numerical values depending on the outcome of a random experiment.
Discrete random variable
A random variable that can take no more than a countable number of values
What is a random variable denoted by?
X
What is P(X=x) or P(x)
The probability that X takes the specific value x
Cumulative distribution function F(x)
Shows the probability that X is less than or equal to x
How can the cumulative distribution function be written as a normal probability function
F(x)= P(X<=x)
Joint probability function
Used to express the probability that X takes the specific value x and simultaneously Y takes the specific value y, as a function of x and y
How is a joint probability function for x and y written
P(x,y) P(X=x n Y=y)
Margin probabilities
Is the probability of one event happening irrespective of another event happening
Conditional probability function
Expresses the probability that X takes the value x when the value y is specified for Y
Equation for conditional probability function
P(x|y) =P(x,y)/P(y)
What is the expected value
The analogous measure of central location for a random variable
Properties of expected values
If X and Y are random variables and b is a constant
1. E(X+Y)= E(X)+E(Y)
2. E(bX)= bE(X)
3. E(b)= b
In general E(g(x)) and g(E(x)) are not equal
Properties of the variance
If V, W and Z are random variables and b is a constant 1. If Y=V+W: Var(Y)= var(V)+var(W)+2cov(V,W) 2.if Y=bZ: Var(Y)=b^2var(Z) 3. If Y=b: Var(Y)=0
What is the covariance when two random variables are statistically independent?
0
Properties of covariance
1. If Y= V+W Cov(X,Y)= cov(X,V)+cov(X,W) 2. If Y=bZ Cov(X,Y)= bcov(X,Z) 3. If Y=b Cov(X,Y)=0
If the covariance is zero what does this tell us?
Nothing. The variables maybe independent or dependent
Permutations
The number of possible oderings with a set of n objects and x ordered boxes
Combinations
We are concerned with the number of different ways that x objects can be selected from n but not concerned about the order
Which is bigger, the permutation or the combination?
Permutation
Bernoulli distribution
A random experiment with only two possible outcomes of
Binomial random variable
The outcome of a series of n independent Bernoulli trials
Poisson distribution
The distribution of the number of times a certain event occurs in a specific time interval or in a specific length or area
Assumptions of Poisson distribution
Assume the interval can be divided into very small sub intervals such that:
- the probability that an event occurs in one sub interval is very small
- the probability of one success in a sub interval is constant for all sub intervals and is proportional to its length
- the sub intervals are independent of each other
What does lambda represent in Poisson distribution?
The mean number for successes in the subinterval and the variance
Differences between Poisson and binomial
- a binomial is limited to the number of trials
- a Poisson can take an infinite number of values
- in binomial mean is greater than variance
- in Poisson mean is equal to variance
When can Poisson distribution be approximated to binomial distribution?
When n is large and p is small
