In a binomial distribution, mean is 3 and standard deviation is `3/2`, then the probability function is

Options

A. `(3/4+1/4)^(12)`
B. `(1/4+3/4)^(12)`
C. `(1/4+3/4)^(9)`
D. `(3/4+1/4)^(9)`

Answer

To find the probability function of a binomial distribution given the mean and standard deviation, we can follow these steps:

### Step 1: Understand the Mean and Standard Deviation in Binomial Distribution
In a binomial distribution, the mean (μ) is given by:
\[ \mu = np \]
And the standard deviation (σ) is given by:
\[ \sigma = \sqrt{npq} \]
where \( q = 1 - p \).

### Step 2: Set Up the Equations
From the problem, we know:
- Mean \( \mu = 3 \)
- Standard deviation \( \sigma = \frac{3}{2} \)

Using the formulas:
1. \( np = 3 \) (1)
2. \( \sqrt{npq} = \frac{3}{2} \) (2)

### Step 3: Square the Standard Deviation Equation
Squaring equation (2):
\[ npq = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

### Step 4: Substitute for \( np \)
From equation (1), we know \( np = 3 \). Substitute this into the squared standard deviation equation:
\[ 3q = \frac{9}{4} \]

### Step 5: Solve for \( q \)
Now, solve for \( q \):
\[ q = \frac{9}{4 \cdot 3} = \frac{9}{12} = \frac{3}{4} \]

### Step 6: Find \( p \)
Since \( p + q = 1 \):
\[ p + \frac{3}{4} = 1 \implies p = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 7: Find \( n \)
Now, substitute \( p \) back into equation (1) to find \( n \):
\[ np = 3 \implies n \cdot \frac{1}{4} = 3 \implies n = 3 \cdot 4 = 12 \]

### Step 8: Write the Probability Function
The probability function for a binomial distribution is given by:
\[ P(X = k) = \binom{n}{k} p^k q^{n-k} \]
Substituting \( n = 12 \), \( p = \frac{1}{4} \), and \( q = \frac{3}{4} \):
\[ P(X = k) = \binom{12}{k} \left(\frac{1}{4}\right)^k \left(\frac{3}{4}\right)^{12-k} \]

### Final Answer
Thus, the probability function is:
\[ P(X = k) = \binom{12}{k} \left(\frac{1}{4}\right)^k \left(\frac{3}{4}\right)^{12-k} \]

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