Random Variables Flashcards
Random Variable:
We call a function X from S to the real values R a random variable if, for every w in S, X(w) is a real number.
Probability distribution of a random variable X
{P[X in B] : is a subset of R}
Extension Theorem
The probability distribution of a random variable X is uniquely determined.
Cumulative distribution function
(cdf)
Let X be a random variable (discrete or continuous).
Let FX(x) be a real-valued function be defined as follows:
FX(x) = P[X <= x]
for all x
Properties of cdf
c) And right continuous
Continuous random variable X
If FX is continuous, then the random variable X is continuous as well.
Example cdf
P[a < X <= b]
FX(b) - FX(b) for all a,b in R
Discrete Random Variable
- cdf is a step function
-
Probability mass function: pX(x) = P[X=x]
- Note: pX(x) = 0 is x cannot be assumed by X
*
- Note: pX(x) = 0 is x cannot be assumed by X
Theorem
The pmf of a discrete random variable uniquely determines the probability distribution of that random variable
How to go from pX(x) to FX(x)?
Remember that pX(x) represents the “jump” height of FX(x) .
So we get pX(x0) = FX(x0) - FX(x0-)
where x0 is fixed.
Basic properties of pX(x)
pX(x) >= 0
Sum over all x that X can assume pX(x) = 1
A plot of pX(x) looks like many points.
Discrete Uniform Distribution
The random variable X has a discrete uniform distribution on the real numbers a1, a2, …, ak
iff
pX(ai) = P[X = ai] = 1/N for all i
Binomial Distributions
The random variable X has a binomial distribution with parameters n and p
iff
P(X=x) = (nCx)(p^x)(1-p)^(n-x)
for x = 0,1,2,…,n
Bernoulli Distribution
The random variable X has a Bernoulli distribution with parameter 0 < p < 1
iff
P(X=x) = p^x(1-p)^(1-x)
for x = 1,0
Bernoulli trials
Only two outcomes, success and failure.
Notice: trials are independent
What do we need to have a binomial distribution?
- We need to have Bernoulli trials
- The trials must be, then, independent
- The probability of a “success” has to be the same for every trial.
Geometric Distribution
“Until success in the x-th trial”
The random variable X has a geometric distribution
iff
P(X=x) = (1-p)^(x-1)p
for x = 1,2,….
Negative Binomial Distribution
“Trials until get x successes”
P(X=x) = (n-1)C(x-1) (1-p)^(n-1-(x-1)) p^(x-1) p
=(n-1)C(x-1) (1-p)^(n-x) p^(x-1) p
Do example in page 36
Posson Distribution
P(X=x) = e^(-lambda) * (lambda^x)/ x!
for x = 0,1,…
Hypergeometric Distribution
Fish-in-the-lake
The random variable X has a hypergeometric distribution with parameters N, n, a
iff
P(X=x) = (aCx)*(N-aCn-x)/ (NCn)
