Week 3 Flashcards
What is a Random Variable?
A function X: Omega -> R that maps an outcome (words) in omega to a number.
Must be defined for all outcomes and single-valued.
When is a random variable discrete/continuous?
Discrete when X(omega) (range of X) is countable.
Continuous when X(omega) is uncountable.
How do you measure the probability of a random variable?
Map back to the sample space.
Find outcome such that X(outcome) = number.
(X^-1(a) is an event in the event space.)
What is the Probability Mass Function?
Px(a) is the probability for the random variable X to take the value a.
The sum of all PMFs for all X(a) should be 1.
Can only describe distribution of discrete random variables.
What is the Cumulative Distribution Function?
FX(x) is the probability for the variable X to be at most x.
Works for discrete, continuous and mixed random variables.
What are the properties of a CMF?
Non-decreasing.
FX(-infinity) = 0 and FX(infinity) = 1.
P(a < X <= b) = FX(b) - FX(a).
What are the Special Distributions?
Bernoulli Distribution: X ~ Bernoulli(p) Binomial Distribution Geometric Distribution: PX(k) = (1-p)^k-1 p (1-p)^k-1 = the first k-1 fails p = last success
Explain the Binomial Distribution?
Pk(x) = (n k) p^k(1-p)^n-k
(n k) = number of combinations
p^k = prob of getting k ps
(1-p)^n-k = prob of getting n-k 1-ps
What is a continuous random variable?
A random variable with an uncountable range X(omega).
Continuous CDF.
What changes when working with continuous random variables?
The probability of any particular number output is 0.
Can only think about intervals.
How do you measure probability of continuous random variables?
Measure the size of a set.
What is the “equiprobable” assumption?
X is equally likely to take any value.
What is the Probability Density Function for continuous random variables?
A function fx: R -> R+, when integrated over interval [a, b], yields probability of obtaining a <= X <= B.
fx(x) is the probability per unit length.
What is a Continuous Uniform Random Variable?
When PDF fx(x) = 1/b-a if a <= x <= b.
CDF = x-a/b-a if a <= x <=b.
What is an Exponential Random Variable?
If a random variable has a memoryless property:
P(X > x+a | X > a) = P (X > x)
fx(x) = \e^-\x, if x >= 0
