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Unformatted text preview: 3.2.5 Negative Binomial Distribution In a sequence of independent Bernoulli( p ) trials, let the random variable X denote the trial at which the r th success occurs, where r is a xed integer. Then P ( X = x | r, p ) = x-1 r-1 p r (1-p ) x-r , x = r, r + 1 , . . . , (1) and we say that X has a negative binomial( r, p ) distribution. The negative binomial distribution is sometimes dened in terms of the random variable Y =number of failures before r th success. This formulation is statistically equivalent to the one given above in terms of X =trial at which the r th success occurs, since Y = X-r . The alternative form of the negative binomial distribution is P ( Y = y ) = r + y-1 y p r (1-p ) y , y = 0 , 1 , . . . . The negative binomial distribution gets its name from the relationship r + y-1 y = (-1) y -r y = (-1) y (-r )(-r-1) (-r-y + 1) ( y )( y-1) (2)(1) , (2) which is the dening equation for binomial coecient with negative integers. Along with which is the dening equation for binomial coecient with negative integers....
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