Let `X ~ B(n,p)` then the mean or expected value of r. v. X is denoted by E(X) and is given by `mu = E(X) = npq .` True or False

Answer

To determine whether the statement "Let `X ~ B(n,p)` then the mean or expected value of r.v. X is denoted by E(X) and is given by `mu = E(X) = npq`" is true or false, we need to analyze the definitions and formulas related to the binomial distribution.

### Step-by-Step Solution:

1. **Understanding the Binomial Distribution**:
- The binomial distribution is defined by two parameters: \( n \) (the number of trials) and \( p \) (the probability of success on each trial). The random variable \( X \) follows a binomial distribution, denoted as \( X \sim B(n, p) \).

2. **Expected Value of a Binomial Distribution**:
- The expected value (mean) of a binomially distributed random variable \( X \) is given by the formula:
\[ E(X) = np \]
- This means that the mean of \( X \) is directly proportional to the number of trials \( n \) and the probability of success \( p \).

3. **Understanding \( q \)**:
- In the context of the binomial distribution, \( q \) is defined as the probability of failure, which can be expressed as:
\[ q = 1 - p \]
- Therefore, the product \( pq \) can be interpreted as \( p(1 - p) \).

4. **Variance of a Binomial Distribution**:
- The variance of a binomially distributed random variable \( X \) is given by:
\[ \text{Var}(X) = npq \]
- This indicates that while the variance involves \( pq \), the expected value does not.

5. **Conclusion**:
- The statement in question claims that the mean \( E(X) \) is equal to \( npq \). However, as established, the mean is actually \( np \) and not \( npq \).
- Therefore, the statement is **False**.

### Final Answer:
**False**