2.11.1 Exponential Flashcards
(12 cards)
what does an exponential distribution model?
The exponential distribution is a continuous probability distribution that is mostly used to model waiting times between independent events that occur with a constant average rate. Examples include the lifetime of a machine, the time between arrival of buses, and the time until the next phone call.
what does the parameter theta mean for the exponential distribution?
This parameter is also the mean of the distribution. When describing the exponential distribution, the question will often state “X is exponentially distributed with mean theta.’’
whats the exponential’s relationship to the poisson?
Poisson random variable models the number of occurrences of an event in a fixed interval. This is related to the exponential distribution, which is used to model the waiting time between events.
whats the relationship between theta from the exponential and lambda from the possion?
theta= 1/lamdba
what is the pdf of the exponential rv parameterized on the mean (θ)?
f(x) = 1/θ * e^-(x/θ) where x>0
what’s the value of θ do to the exponential distribution graph?
generally it’s a exponential decay and theta is the scalar of the shape.
Smaller θ: Steeper decay, events happen quickly.
Larger θ: Slower decay, events take longer on average.
what’s the cdf of the exponential distribution parameterized on the mean (θ)?
F(x) = 1 - e^(-x/θ) where x>0
what’s the survival function of the exponential distribution parameterized on the mean (θ)?
S(x) = e^(-x/θ) where x>0
what is mean and variance of the exponential distribution parameterized on the mean (θ)?
E(X) = θ
Var(X) = θ^2
how does the memoryless property of the exponential r.v. with it’s E(X) and Var(X)?
E[X - c | X > c] = E[X] = E(X) = θ
Var[X - c | X > c] = Var(X) = θ^2
how does the memoryless property of the exponential r.v. with it’s probability?
P(X - c > b |X > c) = P(X > b), where b is greater than 0.
can the exponential be parameterized on (λ) and not θ?
yes.
f(x) = λe^(-λx) where x>0
