[Math] 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…

probabilityprobability distributions

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,

  1. What is the probability distribution table of $X$?

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

  1. 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*}

Best Answer

A good notation for the $CDF$ would be

$$ F_{X}(x)= \begin{cases} 1 & x \geq 15 \\ \frac{4}{6} & 11 \leq x \lt 15 \\ \frac{3}{6} & 10 \leq x \lt 11 \\ \frac{2}{6} & 6 \leq x \lt 10 \\ 0 & x \lt 6 \end{cases} $$

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