# Shared Flashcard Set

## Details

MTH 365 midterm 2 study
covers sections 3.3 to 3.6 of Hogg an Tanis
12
Mathematics
Undergraduate 3
02/24/2014

## Cards Return to Set Details

Term
 moment generating function of a continuous random variable
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
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/(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.
Term
 p.d.f., moment-generating 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 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.
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) ∫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
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
 integration by parts
Definition
 ∫udv = uv -∫vdu
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.
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