The Geometric Probability Distributions

X 為一隨機變數, 若 X 的機率質量函數如最上式所示, 則稱 X 為參數是 p幾何隨機變數(geometric random variable).

Example

Solution:
X 表示取到一個黑球所需的次數, 則 X 滿足 第一式, 其中 p=M/(M+N). 因此
(a)
(b)
Of course, (b) could have been obtained directly, since the probability that at least k trials are necessary to obtain a success is equal to the probability that the first k-1 trials are all failures. That is, for a geometric random variable

Example

Solution
q=1-p, 則我們得

Example

Solution