Cumulative Distribution Function

Example 3.9

I toss a coin twice. Let X be the number of observed heads. Find the CDF of X.

Answer

$X =$ {the number of observed heads}

$X \sim Binomial(2, \frac{1}{2})$, $R_X = \{0, 1, 2\}$.

$$ p_X(k) = \binom{n}{k} p^k (1 - p)^{n - k}, \quad \text{for } k = 0, 1, 2. $$

$$ F_X(x) = P(X \leq x) = \sum_{k \leq x}p_X(k). $$

• $F_X(0) = p_X(0) = 1 * \frac{1}{2}^0 * \frac{1}{2}^{2 - 0} = \frac{1}{4}$
• $F_X(1) = p_X(0) + p_X(1) = \frac{1}{4} +
  = \frac{1}{4}$
• $F_X(2) = 1 * \frac{1}{2}^0 * \frac{1}{2}^{2 - 0} = \frac{1}{4}$