Questions: Construct the discrete probability distribution for the random variable described. Express the probabilities as simplified fractions. The number of tails in 4 tosses of a coin.

Solution Steps

The PMF for the random variable \( X \), representing the number of tails in 4 tosses of a coin, is calculated as follows:

\[ \begin{align_} P(X = 0) & = 0.0625 \\ P(X = 1) & = 0.25 \\ P(X = 2) & = 0.375 \\ P(X = 3) & = 0.25 \\ P(X = 4) & = 0.0625 \\ \end{align_} \]

The mean \( \mu \) of the distribution is calculated using the formula:

\[ \mu = \sum_{k=0}^{n} k \cdot P(X = k) = 0 \times 0.0625 + 1 \times 0.25 + 2 \times 0.375 + 3 \times 0.25 + 4 \times 0.0625 = 2.0 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum_{k=0}^{n} (k - \mu)^2 \cdot P(X = k) = (0 - 2.0)^2 \times 0.0625 + (1 - 2.0)^2 \times 0.25 + (2 - 2.0)^2 \times 0.375 + (3 - 2.0)^2 \times 0.25 + (4 - 2.0)^2 \times 0.0625 = 1.0 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{1.0} = 1.0 \]

Final Answer