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Geometric Distribution

r=1r = 1

X=# of trials until the 1st successX = \text{\# of trials until the 1st success}

xZ+x \in \mathbb{Z}^{+}

P=Probability of SuccessP = \text{Probability of Success}

Probability

P(X=x)P(X=x)

P(X=x)=(1p)x1pP(X=x) = (1-p)^{x-1} p

Where (1p)x1(1-p)^{x-1} is the x1x-1 trials that failed and pp is the xthx^{th} trial that succeeded. Then,

P(X>x)P(X>x)

P(X>x)=j=x+1P(x=j)=j=x+1(1p)x1pP(X>x) = \sum\limits_{j=x+1}^{\infty} P(x=j) = \sum\limits_{j=x+1}^{\infty} (1-p)^{x-1} p

Define q=1pq=1-p

Then

=j=x+1qj1p=pqj=x+1qj=pqqx+11q=qx=\sum\limits_{j=x+1}^{\infty} q^{j-1} p = {p \over q}\sum\limits_{j=x+1}^{\infty} q^{j} = {p \over q} {q^{x+1} \over {1-q}} = q^x

P(Xx)P(X \leq x)

P(Xx)=1P(X>x)=1qxP(X \leq x) = 1 - P(X>x) = 1-q^x

x=1(1p)x1p=p1px=1(1p)x=1\sum\limits_{x=1}^{\infty} (1-p)^{x-1} p = {p \over {1-p}} \sum\limits_{x=1}^{\infty} (1-p)^x = 1