To determine the value of $c$, we need to use the property that the sum of the probabilities for a discrete random variable must equal 1. In this case, the random variable $x$ has a probability mass function (pmf) given by $f(x) = c \cdot x$ for $x = 1, 2, 3, 4$.

That is,

$f(x) = c \cdot x$

To find $c$, we sum the probabilities over all possible values of $x$ and set the sum equal to 1:

$f(1) + f(2) + f(3) + f(4) = 1$

Substitute $f(x) = c \cdot x$ for each value of $x$:

$c \cdot 1 + c \cdot 2 + c \cdot 3 + c \cdot 4 = 1$

$c(1 + 2 + 3 + 4) = 1$

$c(10) = 1$

Solve for $c$:

$c = \frac{1}{10}$

Thus, the value of $c$ is $\frac{1}{10}$.

Do you want more details or have any questions?

Here are 5 related questions to consider:

1. What is the expected value $E(x)$ for this random variable?
2. How would you compute the variance of $x$?
3. If the values of $x$ changed to $x = 2, 4, 6, 8$, how would you adjust the pmf and find the new $c$?
4. How would the pmf change if the range of $x$ included 5 as well?
5. How would you calculate the cumulative distribution function (CDF) for this random variable?

Tip: Always verify that your pmf satisfies the condition that the total probability sums to 1 to ensure correctness.