EX 3.18 expected value of Bernoulli random variable

• Definition of the random variable:

  • Let $X$ be defined as:
    • $X=1$ if a randomly selected vehicle passes the emissions test.
    • $X=0$ if it does not pass.

• Nature of $X$:

  • $X$ is a Bernoulli random variable with the following probability mass function (pmf):
    • $P(X=1)=p$
    • $P(X=0)=1-p$

• Expected value calculation:

  • The expected value of $X$ is computed as follows:
  \begin{align}

E(X) &= 0 \cdot P(X = 0) + 1 \cdot P(X = 1) \ &= 0 \cdot (1 - p) + 1 \cdot p \ &= p \end{align} $$ - Thus, the expected value $E(X)$ is simply the probability that $X$ takes the value 1 - i.e., the probability of passing the test.

• Conceptual population:

  • If we think of a population consisting of:
    • $0$s in proportion $1-p$ (vehicles that fail).
    • $1$s in proportion $p$ (vehicles that pass).
  • The average of this conceptual population is:
  $$\mu =p$$

• Interpretation:

  • The mean $\mu$ represents the overall probability of a vehicle passing the emissions test when considering a large population.