2.4 Statistical Distributions

• Discrete statistical distributions allow values that are specific
• What is the shape of the normal distribution?

  Bell-shaped
• What does the standard deviation measure in a normal distribution?

  Spread of the curve
• The normal distribution is symmetric around its standard deviation.

  False
• What does the standard deviation control in a normal distribution?

  Spread of the curve
• If a student's height is 175 cm, the mean height is 165 cm, and the standard deviation is 10 cm, the Z score is 1.
• Match the distribution type with its example:

  Discrete ↔️ Poisson

  Continuous ↔️ Normal
• Discrete distributions can take on any value within a range.

  False
• The normal distribution is symmetric around its mean.

  True
• In a normal distribution, the mean, median, and mode are all equal
• To transform a value from a normal distribution to a standard normal distribution, we use the Z score
• Approximately 68.2% of data in a normal distribution lies within one standard deviation of the mean.
• Match the property with its description for the binomial distribution:

  Mean ↔️ Equal to \(n \times p\)

  Variance ↔️ Equal to \(n \times p \times (1 - p)\)

  Standard Deviation ↔️ The square root of the variance
• The Poisson distribution is a discrete probability distribution.
• The Poisson distribution is a discrete probability distribution.

  True
• What is the probability of exactly 5 customers arriving in an hour if the average rate is 3 per hour?

  $0.1008$
• What is an example of a discrete distribution?

  Phone calls per day
• Match the properties of the normal distribution with their descriptions:

  Symmetry ↔️ Perfectly symmetric around the mean

  Mean, Median, Mode ↔️ Equal and coincide at the center

  Standard Deviation ↔️ Determines the spread of the distribution
• If a student scores 80 on a test with a mean of 70 and a standard deviation of 5, the Z-score is 2
• Calculations using the normal distribution involve determining probabilities under the normal curve.
• In the binomial distribution, the success probability must remain constant for each trial.
• What are the key conditions for a binomial distribution?

  Fixed trials, independence, constant success probability, two outcomes
• Match the condition with its description:

  Fixed Number of Trials ↔️ A set number of trials

  Independence ↔️ Trials do not affect each other

  Constant Success Probability ↔️ Same probability for each trial

  Two Outcomes ↔️ Each trial can succeed or fail
• The binomial distribution models the probability of \(k\) successes in \(n\) independent trials.

  True
• What is the formula for the variance of a binomial distribution?

  $n \times p \times (1 - p)$
• What is the PMF formula for the Poisson distribution?

  $P(X = k) =$$\frac{e^{ - \lambda} \lambda^{k}}{k!}$
• What is the standard deviation of a Poisson distribution with \(\lambda = 8\)?

  $\sqrt{8}$
• What are the two parameters that define a normal distribution?

  Mean and standard deviation
• What is the formula for calculating the Z-score?

  $Z =$$\frac{x - \mu}{\sigma}$
• Discrete distributions have values that can only take on specific, separate values
• What is another name for the normal distribution?

  Gaussian distribution
• What is the mean of the standard normal distribution?

  0
• Provide an example of a continuous distribution.

  Normal
• The probability mass function (PMF) of the binomial distribution includes the term (1 - p)^(n - k), which represents the probability of failure
• What does the standard deviation measure in the normal distribution?

  Spread of the distribution
• What does the Poisson distribution model?

  Events within a fixed interval
• The Poisson distribution's probability mass function (PMF) includes the term e^{-\lambda}, where λ represents the constant average rate
• What is the base of the natural logarithm in the Poisson distribution's PMF?

  $e \approx 2.71828$
• An example of a discrete statistical distribution is the Poisson
• Heights of students in a school tend to follow a normal distribution with a mean of 165 cm and a standard deviation of 10 cm.