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Conditional probability is a crucial concept in probability theory, allowing us to calculate the likelihood of an event occurring given that another event has already occurred. It's generally denoted as \( \operatorname{Pr}(A \mid B) \), which represents the probability of event \( A \) happening given that \( B \) has already happened.
In the exercise, we deal with binary random variables \( Y_1 \) and \( Y_2 \). Being binary variables means each variable can take one of two possible outcomes, typically 0 or 1. The problem gives us several conditional probabilities:

• \( \operatorname{Pr}(Y_1=1 \mid Y_2=0) = 1 \): If \( Y_2 \) is 0, \( Y_1 \) is 1 with certainty.
• \( \operatorname{Pr}(Y_1=0 \mid Y_2=1) = 1 \): If \( Y_2 \) is 1, \( Y_1 \) must be 0.
• \( \operatorname{Pr}(Y_2=0 \mid Y_1=0) = 1 \): If \( Y_1 \) is 0, \( Y_2 \) must also be 0.
• \( \operatorname{Pr}(Y_2=1 \mid Y_1=1) = 1 \): If \( Y_1 \) is 1, \( Y_2 \) must be 1.

These conditional probabilities suggest that whenever one variable is known, the other is determined completely, showcasing a deterministic relationship, rather than a stochastic one. Understanding these conditional statements helps to identify whether a joint probability distribution can be established.

Binary random variables are a specific type of random variable that take on one of exactly two possible values. These values are often labeled as 0 and 1. This binary nature makes them simple yet powerful tools for modeling various phenomena.
The exercise involves two binary random variables, \( Y_1 \) and \( Y_2 \). The conditional probability relationships given in the problem suggest that these variables have a perfectly deterministic relation: when one is known, the other's state is entirely predictable.
For example:

• If you know \( Y_2=0 \), then \( Y_1 \) must be 1.
• If \( Y_2=1 \), then \( Y_1 \) has to be 0.

These deterministic outcomes signify that any randomness typically expected in joint probability distributions is absent. Instead, it highlights a situation where one variable dictates the value of the other fully. This is contrary to what we usually expect with random variables, where there's some inherent uncertainty involved.

The positivity condition is an essential requirement for forming a valid probability distribution. It demands that all probabilities involved in the distribution must be non-negative and, preferably, positive.
In the provided scenario, we attempted to construct a joint probability distribution based on the given local characteristics. However, the deterministic nature of these relationships led to contradictions. A joint probability distribution must normally allow for simultaneous events (like \( Y_1=1 \) and \( Y_2=1 \)), but here we see contradictions. The variables suggest:

• \( Y_1=1 \) implies \( Y_2=1 \)
• \( Y_2=1 \) implies \( Y_1=0 \)

These conditions cannot occur together, which indicates inconsistency in forming valid joint probabilities. Since we cannot assign a valid positive joint probability to every possible outcome without contradicting another, the positivity condition fails here.
Essentially, the contradictions mean that valid probabilities can't be established for all conditions required, making the joint distribution impossible. Understanding this helps underscore why such strictitude is applied in forming joint probability densities, ensuring they reflect realistic probabilistic situations.