(a) Determine the binomial distribution whose mean is 10 and variance is 8.
(b) Write its probability function.

Answer

To solve the problem step by step, we will determine the binomial distribution whose mean is 10 and variance is 8, and then write its probability function.

### Step 1: Understand the formulas for mean and variance in a binomial distribution
In a binomial distribution, the mean (μ) and variance (σ²) are given by the formulas:
- Mean: \( \mu = np \)
- Variance: \( \sigma^2 = npq \)
where:
- \( n \) = number of trials
- \( p \) = probability of success
- \( q \) = probability of failure (where \( q = 1 - p \))

### Step 2: Set up the equations using the given mean and variance
From the problem, we have:
- Mean \( \mu = 10 \)
- Variance \( \sigma^2 = 8 \)

Using the formulas:
1. \( np = 10 \) (Equation 1)
2. \( npq = 8 \) (Equation 2)

### Step 3: Substitute \( q \) in terms of \( p \)
Since \( q = 1 - p \), we can substitute \( q \) in Equation 2:
\[ np(1 - p) = 8 \]

### Step 4: Substitute \( n \) from Equation 1 into Equation 2
From Equation 1, we can express \( n \) as:
\[ n = \frac{10}{p} \]

Now substitute \( n \) into the modified Equation 2:
\[ \left(\frac{10}{p}\right)p(1 - p) = 8 \]
This simplifies to:
\[ 10(1 - p) = 8 \]

### Step 5: Solve for \( p \)
Now, solve for \( p \):
\[ 10 - 10p = 8 \]
\[ 10p = 2 \]
\[ p = \frac{1}{5} \]

### Step 6: Find \( q \)
Now that we have \( p \), we can find \( q \):
\[ q = 1 - p = 1 - \frac{1}{5} = \frac{4}{5} \]

### Step 7: Substitute \( p \) back to find \( n \)
Using \( p \) in Equation 1 to find \( n \):
\[ n = \frac{10}{p} = \frac{10}{\frac{1}{5}} = 50 \]

### Step 8: Summarize the binomial distribution parameters
We have determined:
- \( n = 50 \)
- \( p = \frac{1}{5} \)
- \( q = \frac{4}{5} \)

### Step 9: Write the probability function
The probability function of a binomial distribution is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
Substituting the values of \( n \), \( p \), and \( q \):
\[ P(X = r) = \binom{50}{r} \left(\frac{1}{5}\right)^r \left(\frac{4}{5}\right)^{50-r} \]

### Final Answers
(a) The binomial distribution is characterized by \( n = 50 \), \( p = \frac{1}{5} \), and \( q = \frac{4}{5} \).

(b) The probability function is:
\[ P(X = r) = \binom{50}{r} \left(\frac{1}{5}\right)^r \left(\frac{4}{5}\right)^{50-r}, \quad r = 0, 1, 2, \ldots, 50 \]