Random Variables and Probability Distributions Flashcards
what is a random variable
A random variable is a function 𝑋 ∶ 𝑆 → R that associates a numerical value X(s) with every outcome s ∈ S of a random experiment
What is the range space
the set of all possible set of values, X can take
A discrete random variable has the property that its range space is a countable set, when is it continuous
A random variable is continuous if its range space is an uncountable set.
What is the cumulative distribution function?
The function that calculates the probability that X will not exceed a given value x.
F(x) = P(X <= x)
In general, what are the the properties of CDF’s?
1) lim F(x) -> - infinity = 0
2) lim F(x) -> + infinity = 0
3) F(x) is a non decreasing Function
Let the random variableX have the cdf F(x) = P(X <= x) and range space R then P(X>x) is
1 - F(x)
Let the random Variable X have the CDF F(x) and range space R, then what is P(X ∈ (a,b])
F(b) - F(a)
What is the probability mass function
for a discrete random variable, with range space R, the probability mass distribution function =
p(x) = {P(X=x) : for all real X’s, 0 for everything else)
What is the probability density function
f(x) = d/dx F(x)
where F(x) = the integral of f(x’) from - infinity to x
What is the probability of finding the random variable X with pdf f(x) in the interval [a,b]
P(a<= x <= b) = integral of f(x) from a - b
What are the properties of the pdf
the sum of P(x) = 1 and p(x) is inbetween 0 and 1
with |𝑅| = 𝑘. Then 𝑋 has the discrete uniform distribution if its probability mass function
is constant, that is
if x is real p(x) = 1/k otherwise p(x) = 0
What is a Bernoulli probability distribution
A discrete random variable, X is said to have a Bernoulli distribution, if its range space consists of two possible outcomes. R = {0,1}
any random experiment that has only two outcomes can be described by a Bernoulli distribution
What is a Bernoulli trial
a random process with two possible outcomes
What is a geometric distribution
P(X = n) = ((1- p)^(n-1))*p
Definition 5.15. The discrete random variable 𝑋 with range space 𝑅 = {1, 2, … } has a geo- metric distribution with probability 𝑝 ∈ (0, 1], written as 𝑋 ∼ Geo(𝑝), if it has probability mass function 𝑝(𝑥) = (1 − 𝑝)𝑥−1𝑝 for 𝑥 ∈ 𝑅 and 𝑝(𝑥) = 0 otherwise.
What is a Poisson distribution?
Definition 5.16. Let 𝑋 be a discrete random variable with range space 𝑅 = {0, 1, 2, … }. We say that 𝑋 has a Poisson distribution with rate parameter 𝜆, written as 𝑋 ∼ Pois(𝜆)
What is an exponential distribution?
Definition 5.17. A continuous random variable 𝑋 with range space 𝑅 = (0,∞) has an exponential distri
