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In probability theory, non-negativity is a fundamental property of probability mass functions (PMFs), including joint PMFs. This simply means that every probability value must be zero or positive. Probabilities are not just numerical values; they represent the likelihood of events occurring. Hence, it doesn't make sense for them to be negative.

Consider our joint PMF example with the pairs \(x, y\) and their probabilities: \

• \((-1.0, -2)\) with \(f_{XY} = \frac{1}{8}\)
• \((-0.5, -1)\) with \(f_{XY} = \frac{1}{4}\)
• \((0.5, 1)\) with \(f_{XY} = \frac{1}{2}\)
• \((1.0, 1)\) with \(f_{XY} = \frac{1}{8}\)

Checking these values, they are all positive or zero, confirming non-negativity. Now, you have a solid footing on how probabilities should behave!

The total probability sum is a key property of a joint PMF, stipulating that the sum of all probabilities across all possible outcomes must equal one. This is a reflection of capturing the entirety of possible events.

Let's take a look at the total probability sum from our example.The probabilities are given as: \[ \frac{1}{8} + \frac{1}{4} + \frac{1}{2} + \frac{1}{8} = 1 \]

• \(\frac{1}{8}\) for \(x=-1.0, y=-2\)
• \(\frac{1}{4}\) for \(x=-0.5, y=-1\)
• \(\frac{1}{2}\) for \(x=0.5, y=1\)
• \(\frac{1}{8}\) for \(x=1.0, y=1\)

Adding these, the sum equals one, confirming that our joint PMF captures all possibilities completely. Remember, a properly defined joint PMF should always sum to one, indicating the certainty of the occurrence of one of the described events.

Calculating the marginal probability distribution involves isolating one variable by summing out the others in the joint PMF. This gives us a probability distribution for a single variable, providing insight into its behavior independent of other variables.

For instance, to find the marginal probability distribution of \(X\), we look at all variables across the fixed \(x\) values, adding them up:

• \(P(X = -1.0) = \frac{1}{8}\)
• \(P(X = -0.5) = \frac{1}{4}\)
• \(P(X = 0.5) = \frac{1}{2}\)
• \(P(X = 1.0) = \frac{1}{8}\)

This distribution shows how variable \(X\) behaves across all outcomes, regardless of \(Y\). Similar calculations can be performed for \(Y\) by summing over all \(x\) values for each \(y\) value.

Expected value and variance are two essential statistical metrics used to describe the behavior of a random variable. The expected value (or mean) represents the average or central value, while the variance measures the spread or how much values deviate from the mean.

Let's see how they're calculated:For \(X\), the expected value \(E(X)\) is given by combining each possible value of \(X\) with its probability:\[ E(X) = (-1.0) \cdot \frac{1}{8} + (-0.5) \cdot \frac{1}{4} + 0.5 \cdot \frac{1}{2} + 1.0 \cdot \frac{1}{8} = -0.125 \] Variance \(V(X)\) is calculated using the formula: \[ V(X) = E(X^2) - (E(X))^2 \] Where \(E(X^2)\) is: \[ E(X^2) = (-1.0)^2 \cdot \frac{1}{8} + (-0.5)^2 \cdot \frac{1}{4} + (0.5)^2 \cdot \frac{1}{2} + (1.0)^2 \cdot \frac{1}{8} = 0.625 \] Thus, \(V(X) = 0.625 - (-0.125)^2 = 0.609375\).

Repeat similar steps for \(Y\):- \(E(Y) = (-2) \cdot \frac{1}{8} + (-1) \cdot \frac{1}{4} + 1 \cdot \frac{1}{2} + 1 \cdot \frac{1}{8} = 0\).- \(V(Y) = 1.875 - 0^2 = 1.875\).These calculations demonstrate how \(X\) and \(Y\) behave in terms of their means and spreads. They are critical for analyzing the characteristics of the variables statistically.