By J.A. White (Auth.)
Publication by way of White, John A., and so forth
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X 2 If two random variables are dependent upon one another, one would expect that the density or probability mass function of one would be in some way dependent upon the value of the other. This dependence is expressed through the conditional density function or the conditional probability mass 42 2. Probability Theory and Transform Methods function. The conditional density function of X given X ,f(*i | x ), is given by x /(xj I x ) 2 = 2 2 ^2) /(*i> and the conditional probability mass function of X given X , p(x \ x ), is defined as 1 2 x 2 Example 2JS The time Τ required to provide a given service is an exponential random variable with parameter A.
1 1-ρ Ρ Ρ π η(1 - ρ) Ρ Ρ (b-a) -γ- -^Γ Μη ~W χ = 1, 2, 3 , . . 2 0<ρ<1 η = 1, 2 , . . b-a + -τ~ *= (Ν - η)Μη(Ν - Μ) (Ν - 0 < ρ< 1 χ = π, η + 1 , . . ,π 2 Ν > M > η Geometric distribution. As in the case of the binomial random variable, the geometric random variable is related to the Bernoulli random variable. Let X be the random variable representing the number of Bernoulli trials necessary to obtain the first one (1). Again, ρ is the probability of obtaining a one on any Bernoulli trial and 1 — ρ is the probability of obtaining a zero.
84) We have already demonstrated the relationship between the Poisson and exponential random variables. There is also an interesting relationship between the Poisson and uniform random variables. Suppose that k Poisson events have occurred in time t. Let the times of occurrence of the first through feth events be v v , v . Let [t , t + dv ], i = 1, 2, k, be nonoverlapping intervals. 85) 50 2. Probability Theory and Transform Methods Hence the joint density function of v v ,.. ·, v givenfcevents in t is given by l9 fc 2 k !
Analysis of Queueing Systems by J.A. White (Auth.)