Module 12: Important Continuous RVs

Question 1

Summarize Continuous Uniform Random Variables

Answer 1

Scanario it models: a variable is equally likely to take on any value within a (time) interval and we want to compute ththe probability the event happened at (more precisely, in a subinterval) or by a certain point in time.

PDF: f(x) = 1/(b - a) when a <= x <= b
0 otherwise

CDF: F(x) = 0 when x < a, (x-a)/(b-a) when a =< x =< b and 1 when x > b

Expectation and Variance: E(X) and median: (a + b)/2 and Var(X) = ((b - a)^2)/12

Question 2

sumarize exponential RV’s

Answer 2

Scenario it models: the time of the future occurences In partilcular, it models the probability of the first occurrence of an event with λ representing the average rate of occurrence.

PDF: f(x) = λe^(-λx) when x >= 0 and 0 otherwise

CDF: F(x) = 0 when x < 0 and 1 - e^(-λx) when x >= 0

Expectiation and Variance: E(X) = 1/λ, Var(X) = 1/(λ^2)

Median: ln(2)/-λ

Question 3

what is the memoryless property?

Answer 3

P({X > s + t} | {X > t}) = P({X > s})

what the eqn says is that probability that an x > s+t will happen given that an X > t has already happened is equal to the probability of X > s happening.

Question 4

what random variables does memoryless property work for?

Answer 4

Exponential and Geometric