Question

Three coins are tossed simultaneously, \( X \) is the number of heads. Find the expected value and variance of \( X \).

Hint:

For a binomial distribution, use the formulas \( E(X) = np \) and \( \text{Var}(X) = np(1 - p) \) to find the expected value and variance.

Answer 1

Step 1: Understand the distribution of \( X \).
Since three coins are tossed simultaneously, the number of heads, \( X \), can take values \( 0, 1, 2, 3 \). The probability distribution of \( X \) is binomial with \( n = 3 \) trials (tosses) and \( p = \frac{1}{2} \) probability of heads. The probability mass function for \( X \) is: \[ P(X = k) = \binom{3}{k} \left( \frac{1}{2} \right)^k \left( \frac{1}{2} \right)^{3-k} = \binom{3}{k} \left( \frac{1}{2} \right)^3 \] For \( k = 0, 1, 2, 3 \), the probabilities are: \[ P(X = 0) = \frac{1}{8}, P(X = 1) = \frac{3}{8}, P(X = 2) = \frac{3}{8}, P(X = 3) = \frac{1}{8} \]

Step 2: Calculate the expected value \( E(X) \).
The expected value of a binomial distribution is given by: \[ E(X) = np = 3 \times \frac{1}{2} = \frac{3}{2} \]

Step 3: Calculate the variance \( \text{Var}(X) \).
The variance of a binomial distribution is given by: \[ \text{Var}(X) = np(1 - p) = 3 \times \frac{1}{2} \times \frac{1}{2} = \frac{3}{4} \]

Final Answer: - Expected value \( E(X) = \boxed{\frac{3}{2}} \) - Variance \( \text{Var}(X) = \boxed{\frac{3}{4}} \)