4.12 The Geometric Distribution

• What type of probability distribution is the geometric distribution?

  Discrete
• In the PMF formula, `k` represents the number of trials
• The variance of the geometric distribution is $\frac{1 - p}{p^{2}}$.


  True
• What is the term for independent trials with only two possible outcomes in a geometric random variable?

  Bernoulli trials
• Match the property with its corresponding formula or description for a geometric random variable:

  PMF ↔️ $P(X = k) =$$(1 - p)^(k - 1) *$$p$

  Mean ↔️ $\frac{1}{p}$

  Variance ↔️ $\frac{1 - p}{p^{2}}$
• What is the probability mass function (PMF) of the geometric distribution?

  $P(X = k) =$$(1 - p)^(k - 1) *$$p$
• What does the term (1 - p)^(k - 1) represent in the PMF of the geometric distribution?

  Probability of failures
• Match the term with its description in the PMF of the geometric distribution:

  X ↔️ Geometric random variable

  k ↔️ Number of trials

  p ↔️ Probability of success
• The geometric random variable X represents the number of trials needed to achieve the first success.


  True
• The term (1 - p)^(k - 1) in the geometric PMF represents the probability of (k - 1) consecutive failures.


  True
• The geometric distribution models the number of trials needed to achieve the first success
• The trials in a geometric distribution are independent Bernoulli trials.


  True
• What is the formula for the probability mass function (PMF) of the geometric distribution?

  $P(X = k) =$$(1 - p)^(k - 1) *$$p$
• What is the mean of the geometric distribution?

  $\frac{1}{p}$
• Match the property with its value for the geometric distribution:

  Mean ↔️ $\frac{1}{p}$

  Variance ↔️ $\frac{1 - p}{p^{2}}$

  PMF ↔️ $P(X = k) =$$(1 - p)^(k - 1) *$$p$
• A geometric random variable represents the number of trials needed to achieve the first success
• The probability of success in each Bernoulli trial of a geometric random variable is denoted as 'p'.


  True
• Steps to identify the characteristics of a geometric random variable:

  1️⃣ Check for independent Bernoulli trials

  2️⃣ Identify the constant probability of success 'p'

  3️⃣ Determine the number of trials until the first success
• The geometric distribution assumes that the trials are independent and each has the same probability of success.


  True
• The geometric distribution is useful for modeling situations where we are interested in the number of trials needed to obtain the first success
• The geometric distribution is a continuous probability distribution.

  False
• Match the property with its formula for the geometric distribution:

  Mean ↔️ $\frac{1}{p}$

  Variance ↔️ $\frac{1 - p}{p^{2}}$
• The probability mass function (PMF) of the geometric distribution is $P(X = k) =$$(1 - p)^(k - 1) *$$p$.


  True
• If you roll a fair die until you get a 6, the number of rolls needed follows a geometric distribution with p = \frac{1}{6}
• What is the probability mass function (PMF) of the geometric distribution?

  $P(X = k) =$$(1 - p)^{(k - 1)} \cdot p$
• Steps to interpret the PMF formula of the geometric distribution:

  1️⃣ Recognize (1 - p)^(k-1) as the probability of (k-1) failures

  2️⃣ Recognize p as the probability of the first success on the k-th trial
• The term 'p' in the geometric distribution represents the probability of success
• What does the term 'k' represent in the geometric distribution?

  Number of trials
• What does the term 'p' in the geometric PMF represent?

  Probability of success on the k-th trial
• The mean of the geometric distribution is 1/p
• What is the variance of the geometric distribution?

  $\frac{1 - p}{p^{2}}$
• What is the probability of obtaining the first success on the 5th trial if the probability of success is 0.2?

  0.1024
• The geometric distribution models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials.
• In Bernoulli trials, there are only two possible outcomes: success or failure.

  True
• What does the geometric random variable 'X' represent?

  Number of trials until first success
• The PMF of the geometric distribution is $P(X = k) =$$(1 - p)^{(k - 1)} \cdot p$


  True
• In the PMF formula, `X` is the geometric random variable representing the number of trials needed to achieve the first success
• The term $(1 - p)^{(k - 1)}$ in the PMF formula represents the probability of (k-1) consecutive failures before the first success.


  True
• The term `p` in the PMF formula represents the probability of the first success on the k-th trial.
• The variance of the geometric distribution is (1 - p) / p^2