Probability Chapter 2 Flashcards
What is a random variable?
A random variable is a function x(s) that associates a unique numerical value x(s) with every outcome S in the sample space S.
What is a discrete random variable?
A discrete random variable x(s) is a random variable whole sample space S only has a countable (finite or countably infinite) number of outcomes
What is a Probability Mass Function?
A PMF p(xi) specifies the probability that the random variable X equals outcome xi for all outcomes xi in the sample space.
p(xi)=P(X=xi)
What is a cumulative distribution function (CDF)?
The CDF represents the probability that a realization from the random variable is less than or equal to r
F(r)P(X<=r)=Sigma(xEs\x<=r) p(x)
What is the expectation?
The expectation of a discrete random variable X with a possibly infinite sample space S & probability mass function is p(x)=P(X=x) is
E(X)= Sigma(xEs) xp(x)
What are the expectation properties?
- E(X+Y)=E(X)+E(Y)
- E(a)=a (expected value of a constant is a constant itself)
- E(aX)=aE(X)
- E(aX+b)=aE(X)+b
If X & Y are independent random variables then
E(XY)=E(X)E(Y)
But this is not true if they are not independent
What is the variance?
The variance of a discrete random variable X is given by
Var(X)=E(X^2)-(E(X))^2
The standard deviation is
σX=(E(X^2)-(E(X))^2)^1/2
What are the variance properties?
- Var(a)=0
- Var(X)>=0
- Var(X)=0 only if x is a constant
- Var(bX)=b^2Var(X)
- Var(bX+a)=b^2Var(X)
If X & Y are independent then
Var(X+Y)=Var(X)+Var(Y)
What is the definition of the Bernoulli distribution?
‘Success’ or ‘fail’ from a single trial (only 2 outcomes)
X~Bern(θ)
What is the definition of the binomial distribution?
No. Of ‘successes’ out of n independent & identical Bernoulli trials
X~Bi(n, θ)
What is the definition of the geometric distribution?
No. Of trials up to & including the first ‘success’
X~Geo(θ)
What is the definition of the negative binomial distribution?
No. Of trials up to & including the kth ‘success’
X~NB(k, θ)
What is the definition of the Poisson distribution?
No. Of ‘events’ that occurr in a fixed unit of time or space
X~Po(μ)
What is the definition of the hypergeometric distribution?
Sampling n elements without replacement from a larger population containing only two types of items
X~HypGeo(n,N,M)
