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A random variable is a fundamental concept in probability theory. It is a variable that takes on values determined by the outcomes of a random phenomenon. In our exercise, the random variable is represented by the letter \(X\), which denotes the number of toys played with by a child. This implies that as each child is observed interacting with toys, \(X\) changes according to their behavior in that specific instance.
Random variables are broadly categorized into two types: discrete and continuous. The variable \(X\) in this exercise is discrete because it only assumes specific values, namely 0, 1, 2, 3, 4, or 5, corresponding to the count of toys.
When working with random variables, it's crucial to establish what each value represents. Here:

• \(X = 0\): No toys played with.
• \(X = 1\): One toy played with.
• \(X = 2\): Two toys played with.
• ...and so on up to \(X = 5\).

Understanding the nature of random variables helps in formulating events, such as determining the probability of playing with certain numbers of toys.

Probability calculation involves determining the likelihood of various outcomes related to random variables. In our exercise, we calculate probabilities for different possible behaviors of children interacting with toys, which is crucial to predict and understand their behavior.

For example, to find the probability of a child playing with at most two toys, we add the probabilities of \(X\) taking values 0, 1, and 2:

• \(P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)\)
• \(P(X \leq 2) = 0.03 + 0.16 + 0.30 = 0.49\)

This adds up to 0.49, meaning there is a 49% chance that a child will play with two or fewer toys.
Similarly, calculating for \(X > 3\), we consider \(X = 4\) or \(X = 5\):

• \(P(X > 3) = P(X = 4) + P(X = 5)\)
• \(P(X > 3) = 0.17 + 0.11 = 0.28\)

This means a 28% likelihood that more than three toys are played with.
Such calculations help derive useful insights and validate hypotheses about behavioral patterns under study.

Discrete distribution is another key concept in probability and statistics. It refers to a probability distribution where the random variable can only take on a finite or countably infinite set of values. In our exercise, the number of toys \(X\) can take values from 0 to 5, forming a discrete distribution.

To understand a discrete distribution, we need a probability mass function (PMF), which provides:

• The probability of obtaining each possible value of the random variable.

In our example, the PMF is given as:

• \(P(X = 0) = 0.03\)
• \(P(X = 1) = 0.16\)
• \(P(X = 2) = 0.30\)
• \(P(X = 3) = 0.23\)
• \(P(X = 4) = 0.17\)
• \(P(X = 5) = 0.11\)

This information allows researchers to predict the frequency or likelihood of the random variable's outcomes.
A deep understanding of discrete distributions is vital as it empowers us to handle any structured uncertainty involving countable scenarios, thus facilitating the decision-making process based on statistical data.