Term
moment generating function
of a continuous random variable |
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Definition
The moment-generating function of a continuous random variable--if it exists--is given by
M(t) = ∫etxf(x)dx, evaluated from -∞ to ∞, -h<t<h |
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Term
| expected value of a continuous random variable |
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Definition
The expected value of a continuous random variable X is given by
μ = E(X) = ∫xf(x)dx evaluated from -∞ to ∞ |
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Term
| p.d.f., moment generating function, mean, and variance of a uniform distribution |
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Definition
f(x)=1/(b-a), a≤x≤b
M(t) = (etb - eta)/t(b - a), t≠0; 0, t=0
μ = (a + b)/2
σ2 = (b - a)2/12
μ and σ2 are relatively easy to determine by finding E(X) and E(X2) using the p.d.f.
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Term
| p.d.f., moment-generating function, mean and variance of an exponential distribution |
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Definition
f(x) = (1/θ)e-x/θ, 0≤x<∞
M(t) = 1/(1 - θt)
μ = M'(t) = θ
σ2 = M''(t) = θ2
These are relatively easy to determine by differentiating the moment-generating function to find E(X) and E(X2).
θ is the mean waiting time for the first change. |
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Term
| probability density function (p.d.f.) of a continuous random variable |
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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) ∫Sf(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
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Term
| distribution function of a continuous random variable |
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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. |
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Term
| product rule for differentiation |
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Definition
| D{f(x)g(x)} = f(x)g'(x) + f'(x)g(x) |
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Term
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Definition
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Term
| What does a gamma distribution tell us? |
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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. |
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Term
| What does an exponential distribution tell us? |
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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. |
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Term
| quotient rule for differentiation |
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Definition
| D{f(x)/g(x)}=[g(x)f'(x)-f(x)g'(x)]/g(x)2 |
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Term
| What does a Poisson distribution tell us? |
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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. |
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