Final answer:
The probability density function for the sum of values of two coins picked at random from the box is discrete, with P(X) = 1/6 for sums of 6 cents or 15 cents, and P(X) = 1/12 for sums of 10 cents or 11 cents.
Step-by-step explanation:
The student has asked to find the probability density function (PDF) of the sum of the values of two coins, selected at random without replacement from a box containing a penny, two nickels, and a dime.
To solve this, we'll first enumerate all possible outcomes of picking two coins. They are (penny & nickel), (penny & nickel), (penny & dime), (nickel & nickel), (nickel & dime), and (nickel & dime). Note that we have accounted for the fact there are two nickels by considering those combinations separately. These combinations lead to sums of 6, 6, 11, 10, 15, and 15 cents respectively.
Next, we calculate the probabilities of each sum. The box has 4 coins, so when picking the first coin, we have 4 options, and for the second coin, we have 3 options, leading to 12 possible outcomes. The probability of each outcome is the number of ways that outcome can occur divided by the total number of outcomes (12).
Thus we have:
- P(X = 6 cents) = 2/12 = 1/6
- P(X = 11 cents) = 1/12
- P(X = 10 cents) = 1/12
- P(X = 15 cents) = 2/12 = 1/6
The resulting discrete probability density function is:
- P(X) = 1/6 when X = 6 cents or X = 15 cents,
- P(X) = 1/12 when X = 10 cents or X = 11 cents.
Moreover, the sum of these probabilities equals 1, which is a necessary condition for any probability density function. Remember that in this case, X represents a discrete random variable since we are talking about distinct outcomes of the sum of the coin values.