2.4.1 Bernoulli Random Variable and its Distribution¶
Definition¶
- Bernoulli Random Variable: A discrete random variable that takes only two possible outcomes, typically coded as 1 (success) and 0 (failure).
- Probability Mass Function (PMF): f(y)=p if y=1 and f(y)=1−p if y=0, where p is the probability of success and q = 1−p is the probability of failure.
Properties¶
Examples¶
- Coin Toss:
- Success (Heads) = 1 with probability p
-
Failure (Tails) = 0 with probability 1-p
-
Sales Call:
- Success (Sale made) = 1 with probability p
- Failure (No sale) = 0 with probability 1-p
- Customer Survey:
- Success (Satisfied) = 1 with probability p
-
Failure (Not satisfied) = 0 with probability 1-p
-
Store Demand:
- Success (Demand > 10 units) = 1 with probability p
- Failure (Demand ≤ 10 units) = 0 with probability 1-p
Application¶
- Die Toss:
Define success for getting a number less than or equal to 2 on a fair six-sided die:
Calculation Example¶
- For a scenario with a probability of success p=0.2:
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