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A discrete random variable is a type of variable that can take on a countable number of distinct values. In probability, these values are often integers, such as 0, 1, 2, 3, and so on. Because the values are countable, you can list them in a sequence. This contrasts with continuous random variables, which can take on an infinite number of values in an interval.
For our exercise, the discrete random variable is the number of keywords in a message. Each message can have 0, 1, 2, 3, or 4 keywords. These are the possible values our random variable can take. The key thing to remember is that with discrete random variables, each value has a specific probability associated with it. This makes calculating and understanding patterns in data more straightforward. It's akin to having a dice where each face represents a number of keywords.

In the context of this problem, messages that arrive at the service center are categorized by the type of message: email or voice. This classification is crucial because each type of message has a different probability distribution for the number of keywords.
Understanding the differences between these message types helps in calculating the overall probability distribution more accurately. For emails, the probability of having 0 to 4 keywords varies significantly compared to voice messages. For example, in the dataset provided, both email and voice messages offer different probabilities for each keyword count. Emails, for instance, are more likely to have three keywords (probability of 0.4) compared to voice messages, which are more likely to have one keyword (probability of 0.4). Knowing this allows us to weigh the probabilities of keywords differently for emails compared to voice messages.

Calculating probabilities involves finding the chance that a particular event occurs. In our problem, we need to find the probability of receiving a message with a specific number of keywords.
To do this, we combine the probabilities of this occurrence for both message types, using their individual probabilities and the overall probability of receiving either an email or a voice message. For a specific number of keywords, we calculate the probability by taking the email probability, multiplying it by the probability of receiving an email, and then adding it to the voice probability multiplied by the probability of receiving a voice message.

• For 0 keywords: \( P(0) = 0.1 \times 0.7 + 0.3 \times 0.3 = 0.16 \)
• For 1 keyword: \( P(1) = 0.1 \times 0.7 + 0.4 \times 0.3 = 0.19 \)
• For 2 keywords: \( P(2) = 0.2 \times 0.7 + 0.2 \times 0.3 = 0.2 \)
• For 3 keywords: \( P(3) = 0.4 \times 0.7 + 0.1 \times 0.3 = 0.31 \)
• For 4 keywords: \( P(4) = 0.2 \times 0.7 + 0 \times 0.3 = 0.14 \)

This method ensures that all possible outcomes are considered.

A probability distribution for a discrete random variable is a listing of all possible values the variable can assume along with their corresponding probabilities. When dealing with discrete variables, we talk about probability mass functions (PMFs), which detail these probabilities.
For our exercise, the probability distribution is shown by the PMF that we've calculated for the number of keywords. Each value of the random variable (number of keywords) has an associated probability:

• 0 keywords: \( P(0) = 0.16 \)
• 1 keyword: \( P(1) = 0.19 \)
• 2 keywords: \( P(2) = 0.2 \)
• 3 keywords: \( P(3) = 0.31 \)
• 4 keywords: \( P(4) = 0.14 \)

These values represent the distribution of probabilities across different outcomes. Furthermore, we verify the distribution by ensuring that the sum of all probabilities equals 1. This confirms that we have considered all possible scenarios in our model. Understanding this distribution allows us to predict the behavior of the messaging system under similar circumstances in the future.