Question 1204803
**1. Define the Random Variable**

* Let X be the number of students who come in for help during your office hours.

**2. Determine the Distribution**

* Since the students arrive at a constant average rate within a given interval (your office hours) and the arrivals are independent, X follows a **Poisson distribution**.

**3. Calculate the Probability of 6 Students**

* The Poisson distribution has a single parameter, λ, which represents the average number of occurrences within the given interval. 
    * In this case, λ = 2 students/hour * 6 hours = 12 students/day

* The probability mass function (PMF) of a Poisson distribution is:
   P(X = k) = (λ^k * e^(-λ)) / k! 
   where:
       * k is the number of occurrences (in this case, the number of students)
       * λ is the average number of occurrences
       * e is the base of the natural logarithm (approximately 2.71828)

* To find the probability of 6 students:
   P(X = 6) = (12^6 * e^(-12)) / 6! 
   P(X = 6) ≈ 0.0113 

   Therefore, the odds of having 6 students come in for help on a particular day are approximately 0.0113.

**4. Calculate Mean, Variance, and Standard Deviation**

* For a Poisson distribution:
    * Mean (μ) = λ = 12 students/day
    * Variance (σ²) = λ = 12 students²/day
    * Standard Deviation (σ) = √λ = √12 ≈ 3.46 students/day

**In summary:**

* The probability of having 6 students come in for help is approximately 0.0113.
* The mean number of students is 12 per day.
* The variance is 12 students²/day.
* The standard deviation is approximately 3.46 students/day.