Math Revision

• What is the expected value of a discrete random variable X denoted by?


  𝐸[𝑋] or 𝜇(𝑋)
• How is the expected value of a discrete random variable X defined?

  As the sum of xk times P(X = xk)
• What is the range of the expected value or mean?


  −∞ to +∞
• What is the expected value when tossing a fair coin where X = 1 for heads?

  $\frac{1}{2}$
• How do you find the expectation of a loaded die with given probabilities?

  Sum of x times pX(x)
• What is the formula for variance of a random variable X?

  $\sigma^2_X =$$E[X] - E[X]^2$
• What is the standard deviation of a random variable X defined as?

  Square root of the variance
• What are the properties of expectation?

  • E[a] = a for constant a
  • E[Y] = a * E[X] + b for Y = a * X + b
• What does descriptive statistics consist of?

  Methods for organizing and summarizing information
• What is the difference between population and sample in statistics?

  Population is the whole group; sample is part
• What are the types of graphical representations in descriptive statistics?

  • Frequency distribution
  • Histogram
  • Pareto Chart
  • Pie chart
  • Horizontal Bar Chart
  • Stem-Leaves Diagram
• What are descriptive measures that indicate the center of a data set called?

  Measures of central tendency
• How is the sample mean calculated?

  Sum of observations divided by number of observations
• What is the mode in a data set?

  Most frequently occurring value
• What is the median in a data set?

  Point that divides the sample in half
• How is the sample variance defined?

  Expectation of squared differences from the mean
• What is the formula for sample standard deviation?

  Square root of the sample variance
• What are quartiles in a data set?

  Values that split the ordered sample into quarters
• What is the range of a sample?

  Q4 - Q0
• What is the Interquartile Range (IQR)?

  Difference between Q3 and Q1
• What is Chebyshev’s rule?

  • At least \(1 - \frac{1}{k^2}\) of observations lie within k standard deviations of the mean.
  • Applies to any quantitative data set.
• What is the 3σ rule for a Gaussian distribution?

  • \(P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 68.27%\)
  • \(P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 95.45%\)
  • \(P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) \approx 99.73%\)
• What is point estimation in inferential statistics?

  Estimating parameters of a probability distribution
• What is the method of maximum likelihood?

  Estimation method for parameters maximizing likelihood
• What is the range for at least 89% of observations?

  Between $\bar{x} - 3s$ and $\bar{x} +$$3s$
• What is the range for at least 75% of observations?

  Between $\bar{x} - 2s$ and $\bar{x} +$$2s$
• What does the 3σ rule state for a Gaussian distribution?

  • $P(\mu - \sigma \leq X \leq \mu + \sigma) \approx 68.27$
  • $P(\mu - 2\sigma \leq X \leq \mu + 2\sigma) \approx 95.45$
  • $P(\mu - 3\sigma \leq X \leq \mu + 3\sigma) \approx 99.73$
• What is the method of estimating parameters of a probability distribution function?

  Point estimation
• What is one method used for point estimation?

  Method of moments
• What is another method used for point estimation?

  Method of maximum likelihood
• What is a hypothesis testing?

  A method for making statistical inferences
• What statistical measures should be calculated for the given data sets?

  • Mean
  • Median
  • Mode(s)
  • Range
  • Sample standard deviation
• What can be concluded from the percentages of observations in relation to Chebyshev's rule?

  • Sample likely from a normal distribution
  • No outliers present in the sample
  • Sample mean and standard deviation are reliable estimates
• What is the axiomatic definition of probability?

  1. Nonnegative: $P(A) > 0$
  2. Unity for certain event: $P(\Omega) =$$1$
  3. Additive over disjoint events
• What is a discrete sample space?

  Countable outcomes in a sample space
• What is an example of a discrete sample space?

  Flipping a coin
• What is an example of a continuous sample space?

  Time duration of a lifetime experiment
• What defines a discrete random variable?

  A function on a discrete sample space
• What is an example of a discrete random variable?

  Number of heads in two coin tosses
• What defines a continuous random variable?

  A function on a continuous sample space