91Ó°ÊÓ

In probability theory, the probability distribution of a random variable describes how probabilities are assigned to different possible outcomes. For independent random variables like \(X\) and \(Y\) in our problem, the probability distribution is a way to understand the potential values \(T = X + Y\) can take and the likelihood of each value.
The probability distribution is determined by:

• Listing all possible outcomes for a given random variable
• Assigning probabilities to each outcome

To construct the probability distribution for \(T\), each pair of possible \(X\) and \(Y\) values is summed. For example, if \(X\) can be 1, 2, or 5 and \(Y\) can be 2 or 4, the sums are 3, 4, 5, 6, 7, and 9. The probability for each \(T\) value is calculated by multiplying the probabilities of the corresponding \(X\) and \(Y\) values and ensuring they add up to 1.

The mean of a random variable, also known as expectation, is a measure of the central tendency of its probability distribution. It provides an average value that \(T\) can assume over many repetitions of the random experiment.
For two independent random variables \(X\) and \(Y\), the mean of their sum \(T = X + Y\) is simply the sum of their means. This can be expressed as:

• \(\mu_T = \mu_X + \mu_Y\)

For the given problem,
\(\mu_X = 2.7\)
\(\mu_Y = 2.6\)
Thus,
\(\mu_T = 2.7 + 2.6 = 5.3\)
The property that means add linearly when combining random variables simplifies many analyses involving independent variables.

Variance is a measure of how much values of a random variable differ from the mean. It provides insight into the spread or variability within the distribution. For the sum of two independent random variables, the variance is the sum of their variances.
Mathematically, this is represented as:

• \(\sigma_T^2 = \sigma_X^2 + \sigma_Y^2\)

In our example:
\(\sigma_X = 1.55\) so \(\sigma_X^2 = 2.4025\)
\(\sigma_Y = 0.917\) so \(\sigma_Y^2 = 0.8419\)
Therefore,
\(\sigma_T^2 = 2.4025 + 0.8419 = 3.2444\)
This shows that the variance of a sum is not merely the sum of standard deviations, but rather involves the squared terms, reflecting the non-linear nature of variance.

Probability tables are a practical tool to present the probability distribution of a random variable concisely. They list each possible outcome a random variable can take and its corresponding probability, providing a clear, organized way to interpret data.
For the random variable \(T\) calculated as \(X + Y\), the probability table might look like this:

• \(T = 3 \Rightarrow P(T=3) = 0.14\)
• \(T = 4 \Rightarrow P(T=4) = 0.35\)
• \(T = 5 \Rightarrow P(T=5) = 0.27\)
• \(T = 6 \Rightarrow P(T=6) = 0.15\)
• \(T = 7 \Rightarrow P(T=7) = 0.06\)
• \(T = 9 \Rightarrow P(T=9) = 0.09\)

Probability tables are easy to read and help verify calculations, ensuring probabilities sum to 1 and rounding up potential math errors in more complex problems.