EX 3.20 interview before job

• Definition of the random variable:

  • Let $X$ be the number of interviews a student has prior to getting a job.

• Probability mass function (pmf):

  • The pmf of $X$ is given by:
  $$p(x)=\{\begin{array}{ll}k/x^2& \text{for }x=1,2,3,\dots \\ 0& \text{otherwise}\end{array}$$
  • Here, $k=\frac{{\pi }^2}{6}$ is a normalization constant ensuring that the total probability sums to 1:
  $$\sum _{x=1}^{\infty }p(x)=1$$

• Expected value of $X$:

  • The expected value $\mu$ is calculated as follows:
  $$\mu =E(X)=\sum _{x=1}^{\infty }x\cdot p(x)=\sum _{x=1}^{\infty }x\cdot \frac{k}{x^2}$$
  • Simplifying the expression:
  $$\mu =k\sum _{x=1}^{\infty }\frac{1}{x}$$

• Important note:

  • The series ${\sum }_{x=1}^{\infty }\frac{1}{x}$ diverges, indicating that the expected value $\mu$ is infinite.

• Conclusion:

  • The interpretation of $\mu$ shows that, on average, a student is expected to have an infinite number of interviews before getting a job, reflecting a potentially very high number of trials in this scenario.

• Harmonic series:

  • The sum in Equation (3.10):
  $$\sum _{x=1}^{\infty }\frac{1}{x}$$

  is known as the harmonic series.

  • It is a famous series in mathematics that diverges to infinity:
  $$\sum _{x=1}^{\infty }\frac{1}{x}=\infty$$

• Expected value implications:

  • Since $E(X)$ is based on this divergent series, it follows that:
  $$E(X)\text{ is not finite.}$$
  • This occurs because the pmf $p(x)$ does not decrease sufficiently fast as $x$ increases.

• Heavy tails:

  • The distribution of $X$ is said to have “a heavy tail.”
  • Heavy tails refer to probability distributions that allocate a significant amount of probability to values far from the mean $\mu$.

• Consequences of heavy tails:

  • When a sequence of values of $X$ is drawn from this distribution, the sample average will not converge to a finite number; instead, it will tend to grow without bound.

• General implications in statistics:

  • The term “heavy tails” can apply to any distribution where a considerable amount of probability lies far from the mean, not necessarily implying that $\mu =\infty$.
  • Heavy tails complicate statistical inferences about the mean $\mu$, making it challenging to draw conclusions about the expected outcomes.

• Conclusion:

  • Understanding the nature of heavy tails is crucial for interpreting distributions and making reliable statistical inferences.