A discrete random variable 'X' has mean equal to 3 and variance equal to 2. Assuming that the underlying distribution of 'X' is binomial, find the distribution and hence obtain :
(i) P(X = 0)
(ii) Draw a histogram for the distribution.

Answer

To solve the problem, we need to find the parameters of the binomial distribution given the mean and variance of the random variable \(X\).

### Step 1: Identify the Mean and Variance Formulas
In a binomial distribution, the mean (\(\mu\)) and variance (\(\sigma^2\)) are given by:
- 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
From the problem, we know:
- Mean (\(\mu\)) = 3
- Variance (\(\sigma^2\)) = 2

This gives us two equations:
1. \(np = 3\) (1)
2. \(npq = 2\) (2)

### Step 3: Substitute \(q\) in Terms of \(p\)
Since \(q = 1 - p\), we can substitute \(q\) in equation (2):
\[ np(1 - p) = 2 \]
Substituting \(np = 3\) from equation (1):
\[ 3(1 - p) = 2 \]

### Step 4: Solve for \(p\)
Now, we can solve for \(p\):
\[ 3 - 3p = 2 \\ 3p = 1 \\ p = \frac{1}{3} \]

### Step 5: Find \(q\)
Now that we have \(p\), we can find \(q\):
\[ q = 1 - p = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 6: Find \(n\)
Now, we can substitute \(p\) back into equation (1) to find \(n\):
\[ n \cdot \frac{1}{3} = 3 \\ n = 3 \cdot 3 = 9 \]

### Step 7: Summary of Parameters
We have found:
- \(n = 9\)
- \(p = \frac{1}{3}\)
- \(q = \frac{2}{3}\)

Thus, the binomial distribution is \(X \sim B(9, \frac{1}{3})\).

### Step 8: Calculate \(P(X = 0)\)
Using the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]
For \(X = 0\):
\[ P(X = 0) = \binom{9}{0} \left(\frac{1}{3}\right)^0 \left(\frac{2}{3}\right)^{9} \\ = 1 \cdot 1 \cdot \left(\frac{2}{3}\right)^{9} \\ = \left(\frac{2}{3}\right)^{9} \]

Calculating \(\left(\frac{2}{3}\right)^{9}\):
\[ = \frac{2^9}{3^9} = \frac{512}{19683} \]

### Step 9: Draw a Histogram
To draw a histogram for the distribution, we would plot the probabilities \(P(X = r)\) for \(r = 0, 1, 2, \ldots, 9\).

1. Calculate \(P(X = r)\) for \(r = 0, 1, 2, \ldots, 9\) using the binomial formula.
2. Use a graphing tool or software to plot the values.