Geometric Distributionr=1r = 1r=1 X=# of trials until the 1st successX = \text{\# of trials until the 1st success}X=# of trials until the 1st success x∈Z+x \in \mathbb{Z}^{+}x∈Z+ P=Probability of SuccessP = \text{Probability of Success}P=Probability of Success Probability P(X=x)P(X=x)P(X=x) P(X=x)=(1−p)x−1pP(X=x) = (1-p)^{x-1} pP(X=x)=(1−p)x−1p Where (1−p)x−1(1-p)^{x-1}(1−p)x−1 is the x−1x-1x−1 trials that failed and ppp is the xthx^{th}xth trial that succeeded. Then, P(X>x)P(X>x)P(X>x) P(X>x)=∑j=x+1∞P(x=j)=∑j=x+1∞(1−p)x−1pP(X>x) = \sum\limits_{j=x+1}^{\infty} P(x=j) = \sum\limits_{j=x+1}^{\infty} (1-p)^{x-1} pP(X>x)=j=x+1∑∞P(x=j)=j=x+1∑∞(1−p)x−1p Define q=1−pq=1-pq=1−p Then =∑j=x+1∞qj−1p=pq∑j=x+1∞qj=pqqx+11−q=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=j=x+1∑∞qj−1p=qpj=x+1∑∞qj=qp1−qqx+1=qx P(X≤x)P(X \leq x)P(X≤x) P(X≤x)=1−P(X>x)=1−qxP(X \leq x) = 1 - P(X>x) = 1-q^xP(X≤x)=1−P(X>x)=1−qx ∑x=1∞(1−p)x−1p=p1−p∑x=1∞(1−p)x=1\sum\limits_{x=1}^{\infty} (1-p)^{x-1} p = {p \over {1-p}} \sum\limits_{x=1}^{\infty} (1-p)^x = 1x=1∑∞(1−p)x−1p=1−ppx=1∑∞(1−p)x=1인사가 만사다Previous PageAWS Edge ContinuumNext Page