Probability Mass Function For each possible value x of X: 0 Collect all the possible outcomes that give rise to the event =x). Add their probabilities to obtain px(x). px(x) Sample space 2 Event {=x}
Probability Mass Function ◼ For each possible value 𝑥 of 𝑋: ❑ Collect all the possible outcomes that give rise to the event 𝑋 = 𝑥 . ❑ Add their probabilities to obtain 𝑝𝑋(𝑥). Event 𝑋 = 𝑥 Sample space Ω 𝑝𝑋(𝑥)
Important specific distributions Binomial random variable Geometric random variable Poisson random variable
Important specific distributions ◼ Binomial random variable ◼ Geometric random variable ◼ Poisson random variable
Bernoulli Random Variable -The Bernoulli random variable takes the two values 1 and 0 X∈{0,1} ■Its PMF is p=日-p ifx=1 ifx=0
Bernoulli Random Variable ◼ The Bernoulli random variable takes the two values 1 and 0 𝑋 ∈ 0,1 ◼ Its PMF is 𝑝𝑋 𝑥 = ቊ 𝑝 if 𝑥 = 1 1 − 𝑝 if 𝑥 = 0
Example of Bernoulli Random Variable The state of a telephone at a given time that can be either free or busy. A person who can be either healthy or sick with a certain disease. The preference of a person who can be either for or against a certain political candidate
Example of Bernoulli Random Variable ◼ The state of a telephone at a given time that can be either free or busy. ◼ A person who can be either healthy or sick with a certain disease. ◼ The preference of a person who can be either for or against a certain political candidate
The Binomial Random Variable A biased coin is tossed n times. Each toss is independently of prior tosses Head with probability p. Tail with probability 1-p. The number X of heads up is a binomial random variable
The Binomial Random Variable ◼ A biased coin is tossed 𝑛 times. ◼ Each toss is independently of prior tosses ❑ Head with probability 𝑝. ❑ Tail with probability 1 − 𝑝. ◼ The number 𝑋 of heads up is a binomial random variable