<s>[INST] A box contains a penny, two nickels, and a dime. If two coins are selected randomly from the box, without replacement, and if X is the sum... A box contains a penny (1¢), two nickels (5¢), and a dime (10¢). If two coins are selected randomly from the box, without replacement, and if $X$ is the sum of the values of the two coins,


*

*What is the probability distribution table of $X$?


$$\begin{array}{|c|c|c|c|c|}\hline X & 6¢ & 10¢ & 11¢ & 15¢ \\ \hline f(x) & 2/6 & 1/6 & 1/6 & 2/6\\\hline\end{array}$$  


*What is the cumulative distribution function $F(x)$ of $X$?

The cumulative distribution function, $F(x)$ of $X$ is defined as:   $F(x) = P(X ≤ x)$  

So would that mean I just write:
$P(X ≤ 6) = 2/6$
$P(X ≤ 10) = 1/6$
$P(X ≤ 11) = 1/6$
$P(X ≤ 15) = 2/6$  
\begin{align*}
P(X \leq 6) & = P(X = 6)=2/6\\
P(X \leq 10) & = P(X = 6) + P(X = 10)=2/6+1/6=1/2\\
P(X \leq 11) & = P(X = 6) + P(X = 10) + P(X = 11)=1/2+1/6=2/3\\
P(X \leq 15) & = P(X = 6) + P(X = 10) + P(X = 11) + P(X = 15)=2/3+2/6=1
\end{align*}
 [/INST] A good notation for the $CDF$ would be
$$ F_{X}(x)=  
\begin{cases} 
1 & x \geq 15 \\
\frac{4}{6} & 11 \leq x \lt 15 \\
\frac{3}{6} & 10 \leq x \lt 11 \\
\frac{2}{6} & 6 \leq x \lt 10 \\
0 & x \lt 6
\end{cases} $$  
</s>