3.2.1 A Parameter of a Probability Distribution

• PMF of Bernoulli Random Variable $X$:
  • EX 3.9 laptop or desktop
    • $p(0)=0.8$
    • $p(1)=0.2$
      • Reason: $20\%$ of purchasers selected a desktop computer
• Alternative Store Scenario:
  • $p(0)=0.9$
  • $p(1)=0.1$
• General Form of PMF:
  • $p(1)=\alpha$
  • $p(0)=1-\alpha$
    • Condition: $0<\alpha <1$
  • Notation:
    • PMF often written as $p(x;\alpha )$ instead of just $p(x)$ p\left( {x;\alpha }\right) = \left\{ \begin{matrix} 1 - \alpha & \text{ if }x = 0 \\ \alpha & \text{ if }x = 1 \\ 0 & \text{ otherwise } \end{matrix}\right. \tag{3.1}
• Then each choice of $\alpha$ in Expression (3.1) yields a different pmf.

parameters of distributions

• Parameter of a distribution:
  • A quantity that influences the probability distribution $p(x)$.
  • Each value of the parameter results in a different probability distribution.
• Family of probability distributions:
  • The collection of all probability distributions corresponding to different values of the parameter.

• Parameter $\alpha$ in Expression (3.1):
  • Represents a parameter for the Bernoulli distribution.
• Each value of $\alpha$ (where $0<\alpha <1$) corresponds to:
  • A different member of the Bernoulli family of distributions.

EX 3.12 gender of newborn child