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}

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

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

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

• $F_X(0) = p_X(0 \leq x < 1) = p_X(0) = 1 \times \frac{1}{2}^0 \times \frac{1}{2}^{2 - 0} = \frac{1}{4}$
• $F_X(1) = p_X(1 \leq x < 2) = p_X(0) = p_X(0) + p_X(1) = \frac{1}{4} + 2 \times \frac{1}{2}^1 \times \frac{1}{2}^{2 - 1} = \frac{3}{4}$
• $F_X(2) = p_X(x \geq 2) = p_X(0) = p_X(0) + p_X(1) + p_X(2) = \frac{3}{4} + 1 \times \frac{1}{2}^2 + \frac{1}{2}^{2 - 2} = 1$

$$ F_X(x) = \begin{cases} 0, & \text{for } x < 0 \\ \frac{1}{4}, & \text{for } 0 \leq x < 1 \\ \frac{3}{4}, & \text{for } 1 \leq x < 2 \\ 1, & \text{for } x \geq 2 \end{cases} $$