Consider the following statements related to a veriable X having a binomial distribution `b_(x)(n,p)`
1. If `p = 1/2`, then the distribution is symmetrical.
2. p remaining constant, P(X = r) increases as n increases. Which of the statements given above is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

To determine the correctness of the statements related to a variable \( X \) having a binomial distribution \( B(n, p) \), we will analyze each statement step by step.

### Step 1: Analyze the first statement
**Statement 1**: If \( p = \frac{1}{2} \), then the distribution is symmetrical.

- A binomial distribution \( B(n, p) \) is symmetrical when \( p = \frac{1}{2} \). This is because when \( p = \frac{1}{2} \), the probabilities of success and failure are equal, leading to a symmetric distribution around the mean \( \frac{n}{2} \).
- Therefore, **Statement 1 is correct**.

### Step 2: Analyze the second statement
**Statement 2**: \( p \) remaining constant, \( P(X = r) \) increases as \( n \) increases.

- The probability mass function for a binomial distribution is given by:
\[ P(X = r) = \binom{n}{r} p^r (1-p)^{n-r} \]
- If \( p \) is constant and we increase \( n \), the term \( \binom{n}{r} \) (which represents the number of ways to choose \( r \) successes from \( n \) trials) increases. This can lead to an increase in \( P(X = r) \) for fixed \( r \) as \( n \) increases.
- Thus, **Statement 2 is also correct**.

### Conclusion
Both statements are correct.

### Final Answer
Both statements given above are correct.

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