The Binomial Random Variable We refer to X as a binomial random variable with parameters n and p. For k =0,1,...,n. ex(k)=P(X=k)=()p*(1-p)-k
The Binomial Random Variable We refer to 𝑋 as a binomial random variable with parameters 𝑛 and 𝑝. For 𝑘 = 0,1, … , 𝑛. 𝑝𝑋 𝑘 = 𝑃 𝑋 = 𝑘 = 𝑛 𝑘 𝑝 𝑘 1 − 𝑝 𝑛−𝑘
The Binomial Random Variable Normalization （份)p(1-m)=1 k=0 Px(k) Px(k) Binomial PMF n 9,p=1/2 Binomial PMF n Large,p=Small l山于 123456789k 0 n
The Binomial Random Variable Normalization 𝑘=0 𝑛 𝑛 𝑘 𝑝 𝑘 1 − 𝑝 𝑛−𝑘 = 1
The Geometric Random Variable Independently and repeatedly toss a biased coin with probability of a head p,where 0< p<1. The geometric random variable is the number X of tosses needed for a head to come up for the first time
The Geometric Random Variable Independently and repeatedly toss a biased coin with probability of a head 𝑝, where 0 < 𝑝 < 1. The geometric random variable is the number 𝑋 of tosses needed for a head to come up for the first time
The Geometric Random Variable The PMF of a geometric random variable px(k)=(1-p)k-1p k-1 tails followed by a head. Normalization condition is satisfied: 00 ∑pz)=∑1-p)-=p∑(1-p) k= k=1 k=0 1 =p1--0=1
The Geometric Random Variable The PMF of a geometric random variable 𝑝𝑋 𝑘 = 1 − 𝑝 𝑘−1𝑝 𝑘 − 1 tails followed by a head. Normalization condition is satisfied: 𝑘=1 ∞ 𝑝𝑋 𝑘 = 𝑘=1 ∞ 1 − 𝑝 𝑘−1𝑝 = 𝑝 𝑘=0 ∞ 1 − 𝑝 𝑘 = 𝑝 ⋅ 1 1− 1−𝑝 = 1
The Geometric Random Variable The px(k)=(1-p)k-Ip decreases as a geometric progression with parameter 1-p. px(k) p 0123
The Geometric Random Variable The 𝑝𝑋 𝑘 = 1 − 𝑝 𝑘−1𝑝 decreases as a geometric progression with parameter 1 − 𝑝