Term
moment generating function of a continuous random variable 

Definition
The momentgenerating function of a continuous random variableif it existsis given by
M(t) = ∫e^{tx}f(x)dx, evaluated from ∞ to ∞, h<t<h 


Term
expected value of a continuous random variable 

Definition
The expected value of a continuous random variable X is given by
μ = E(X) = ∫xf(x)dx evaluated from ∞ to ∞ 


Term
p.d.f., moment generating function, mean, and variance of a uniform distribution 

Definition
f(x)=1/(ba), a≤x≤b
M(t) = (e^{tb } e^{ta})/t(b  a), t≠0; 0, t=0
μ = (a + b)/2
σ^{2 }= (b  a)^{2}/12
μ and σ^{2 }are relatively easy to determine by finding E(X) and E(X^{2}) using the p.d.f.



Term
p.d.f., momentgenerating function, mean and variance of an exponential distribution 

Definition
f(x) = (1/θ)e^{x/θ}, 0≤x<∞
M(t) = 1/(1  θt)
μ = M'(t) = θ
σ^{2 }= M''(t) = θ^{2}
These are relatively easy to determine by differentiating the momentgenerating function to find E(X) and E(X^{2}).
θ is the mean waiting time for the first change. 


Term
probability density function (p.d.f.) of a continuous random variable 

Definition
The probability density function (p.d.f.) of a random variable X of the continuous type satisfies the folllowing conditions:
(a) f(x) > 0, x ε S
(b) ∫_{S}f(x)dx=1
(c) If (a,b) is a subset of S, then the probability of the event {a < X < b} is P(a < X , B) = ∫f(x)dx evaluated from a to b



Term
distribution function of a continuous random variable 

Definition
The distribution function of a random variable X of the continuous type is given by
F(x)=P(X≤x)=∫f(t)dt evaluated from ∞ to x. 


Term
product rule for differentiation 

Definition
D{f(x)g(x)} = f(x)g'(x) + f'(x)g(x) 


Term

Definition


Term
What does a gamma distribution tell us? 

Definition
A gamma distribution can tell us the probability associated with the waiting time X for a certain number of changes to occur in a Poisson process.
The parameter α (alpha) represents the number of changes we are interested in observing and θ (theta) represents the mean waiting "time" between changes. I write "time" in quotes because we could be talking about changes per foot, etc. as well. 


Term
What does an exponential distribution tell us? 

Definition
The exponential distribution can tell us the probability associated with the waiting time X for a the first change to occur in a Poisson process.
The parameter θ represents the mean waiting "time" for the first change. I write "time" in quotes because we could be talking about changes per foot, etc. as well. 


Term
quotient rule for differentiation 

Definition
D{f(x)/g(x)}=[g(x)f'(x)f(x)g'(x)]/g(x)^{2} 


Term
What does a Poisson distribution tell us? 

Definition
The Poisson distribution can tell us the probability associated with the number of changes X occuring during a period of time.
The parameter λ represents the mean number of changes per period of "time." I write "time" in quotes because we could be talking about changes per foot, etc. as well. 

