Update 1.md

_problems/unit-02/D-cdf-of-a-discrete-RV/1.md

@@ -1,5 +1,48 @@
 ---
 index: 1
 statement: |
-    Problem needed.
----
+    Let X :== as a Random Variable representing the number of heads in four independent flips of a fair coin.
+
+    1. Provide in bracket notation the pmf of X;
+    2. Compute the probability that X is equal to an odd number;
+    3. Provide in bracket notion the cdf of X
+
+Problem modified from Hogg, McKean and Craig - Introduction to Mathematical Statistics     
+---
+Let $Y$ be a bernoulli random variable defined such that each flip
+
+$$   
+Y= \\{ T, H \\} :== \\{ 0, 1 \\} \text{, where T :== Tails and H :== Heads}
+$$
+
+
+$$
+P(Y = y) = 
+\begin{cases} 
+0.5, &y = 0 \\
+0.5, &y = 1 \\
+0, & \text{otherwise} 
+\end{cases}
+$$
+
+Let $X$ be a random variable =  number of Heads (defined as 1's) in four independent flips of a fair coin, e.g. four bernoulli trials
+
+$$ 
+X = \sum_{i=1}^{4} Y(i)\
+$$
+
+$$
+\Rightarrow \text{ X can take on the values } \\{0, 1, 2, 3, 4\\} \text{ heads.}
+$$
+
+To construct the pdf for $X$ let's compute the probability for possible value of $X = x$
+
+<div align="center">
+
+| Number of H, X=x |  P(X=x) as formula    | P(X=x) as value |
+|:----------------:|:---------------------:|:---------------:|
+|        0         |$$\( \binom{4}{0} \times 0.5^0 \times 0.5^4 \)$$|   0.0625    |
+|        1         |$$\( \binom{4}{1} \times 0.5^1 \times 0.5^3 \)$$|   0.25      |
+
+
+</div>