Chapter 4

Distributions of Random Variables

Learning Outcomes

• Define the standardized (Z) score of a data point as the number of standard deviations it is away from the mean: $Z = \frac{x - \mu}{\sigma}$.
• Use the Z score
  • if the distribution is normal: to determine the percentile score of a data point (using technology or normal probability tables)
  • regardless of the shape of the distribution: to assess whether or not the particular observation is considered to be unusual (more than 2 standard deviations away from the mean)
• Depending on the shape of the distribution determine whether the median would have a negative, positive, or 0 Z score.
• Assess whether or not a distribution is nearly normal using the 68-95-99.7% rule or graphical methods such as a normal probability plot.
  • Reading: Section 4.1 of OpenIntro Statistics
  • Test yourself: True/False: In a right skewed distribution the Z score of the median is positive.
• If X is a random variable that takes the value 1 with probability of success $p$ and 0 with probability of success $1-p$, then $X$ is a Bernoulli random variable.
• The geometric distribution is used to describe how many trials it takes to observe a success.
• Define the probability of finding the first success in the $n^{th}$ trial as $(1-p)^{n-1}p$.
  • $\mu = \frac{1}{p}$
  • $\sigma^2 = \frac{1-p}{p^2}$
  • $\sigma = \sqrt{\frac{1-p}{p^2}}$
• Determine if a random variable is binomial using the four conditions:
  • The trials are independent.
  • The number of trials, n, is fixed.
  • Each trial outcome can be classified as a success or failure.
  • The probability of a success, p, is the same for each trial.
• Calculate the number of possible scenarios for obtaining $k$ successes in $n$ trials using the choose function: ${n \choose k} = \frac{n!}{k!~(n - k)!}$.
• Calculate probability of a given number of successes in a given number of trials using the binomial distribution: $P(k = K) = \frac{n!}{k!~(n - k)!}~p^k~(1-p)^{(n - k)}$.
• Calculate the expected number of successes in a given number of binomial trials $(\mu = np)$ and its standard deviation $(\sigma = \sqrt{np(1-p)})$.
• When number of trials is sufficiently large ($np \ge 10$ and $n(1-p) \ge 10$), use normal approximation to calculate binomial probabilities, and explain why this approach works.

Supplemental Readings

• OpenIntro Statistics slides

Videos