91Ó°ÊÓ

Uniform distribution is a fundamental concept in probability where every outcome in the range is equally likely to happen. In this case, we deal with two linked uniform distributions.

• The random parameter, \( A \), follows a uniform distribution over \((0, 1)\).
• Given any specific \( a \) from this range, the variable \( X \) is uniformly distributed over \((-a, a)\), meaning it is equally likely to be anywhere within this interval.

A practical way to view this is imagining the spread of values \( X \) can take depends on the value of \( A \).
This makes \( X \) defined over a constantly shifting interval, depending on \( A \). This two-step definition defines \( X \)'s behavior based on \( A \)'s value, making the analysis a bit more layered.

The expected value is a measure of central tendency, similar to the "average" idea, but in the context of probability. For a uniform distribution on a symmetric interval \((-a, a)\), such as in the scenario for \( X \) given \( A = a \), the expected value is zero.
This stems from symmetry; since the distribution is evenly spread out over \((-a, a)\), all positive outcomes cancel out the negative ones.
For the exercise, the expected value of \( X \) overall is also zero because when \( X \) is evaluated over all possible \( A \), the symmetrical nature persists:

• \( E[X|A = a] = 0 \)
• Thus, \( E[X] = E[E[X|A]] = 0 \)

This gives a quick insight into \( X \)'s behavior without even having to calculate directly.

Variance measures how spread out the values are around the expected value. For \( X \), given any \( A = a \), the variance is \( \frac{a^2}{3} \), characterizing how the values of \( X \) disperse within \((-a, a)\).
To find the overall variance of \( X \), the law of total variance is used:

• \( \operatorname{Var}(X) = E[\operatorname{Var}(X|A)] + \operatorname{Var}(E[X|A]) \)
• \( \operatorname{Var}(E[X|A]) = 0 \)
• \( E[\operatorname{Var}(X|A)] = \int_0^1 \frac{a^2}{3} \, da = \frac{1}{9} \)

This calculation reflects how, despite the dependencies on \( A \), there is a consistent measure of spread in \( X \)'s values.

Conditional probability helps understand the probability of an event, given that another event has occurred. Here, we deal with the conditional distribution of \( X \) given \( A = a \).

• This is expressed as \( f_{X|A}(x|a) = \frac{1}{2a} \), showing the probability density function of \( X \) over \((-a, a)\).
• It informs us exactly how \( X \) behaves when a specific \( A \) is known, illustrating the dependency of \( X \) on \( A \).

Such relationships are fundamental in transitioning from information about one part of the system (\( A \)) to the behavior of another (\( X \)).
By integrating these conditional probabilities over all possible values of \( a \), we derived the marginal distribution \( f_X(x) \). This integrates the dependency of \( X \) on \( A \) into a more global understanding, yielding \( f_X(x) = -\ln|x| \) on \((-1, 1)\).
Understanding this conditional structure is key to analyzing complex systems where outcomes depend on underlying factors.