"What distribution do you think this is?"
Aoi showed Yuki a graph. A bell-shaped curve.
"Normal distribution?"
"Correct. But have you thought about why normal distribution is so important?"
Yuki shook their head.
"Because of the central limit theorem. The sum of independent random variables approaches normal distribution."
Riku entered the club room. "Probability talk again?"
"What else would we talk about?" Aoi laughed.
"Normal chat..."
"Probability is chat. The world is made of probability."
Yuki showed interest. "The world is probability?"
"Quantum mechanically, literally so. But in information theory too, probability distributions are the foundation of everything."
Aoi drew multiple graphs. Normal distribution, exponential distribution, Bernoulli distribution.
"Each a different probability distribution. But they have common properties."
"Expected value?" Yuki said.
"Yes. E[X] = Σ x p(x). Represents the 'center' of a probability distribution."
Riku thought. "Like average?"
"Precisely, probabilistic average. The value obtained on average over many trials."
"Is high expected value good?" Yuki asked.
"Depends on context. Lottery expected value is negative, but people buy."
"Why?"
"Because not just expected value, but also variance matters. Variance represents spread."
Aoi wrote an equation. "Var(X) = E[(X - E[X])²]"
"Mean of squared deviations from expected value."
Riku raised his hand. "Lottery has low expected value but high variance?"
"Exactly. Small probability, large profit. This attracts people."
Yuki understood. "Risk and return."
"Information theory and economics intersect here. Both handle probability distributions."
Riku asked. "Is there an optimal probability distribution?"
"Depends on objective. To maximize entropy, uniform distribution."
"Uniform distribution?"
"All events equally probable. This has highest uncertainty."
Aoi drew a new diagram. Flat distribution.
"But with constraints, it's different. For example, when mean is fixed, the distribution that maximizes entropy is normal distribution."
Yuki was surprised. "Normal distribution has maximum entropy?"
"When mean and variance are fixed, yes. This is one reason normal distribution appears so often."
"Nature maximizes entropy?"
"In a sense. With little information, the least biased distribution appears."
Riku asked another question. "What's moment generating function?"
Aoi was surprised. "Where did you hear that?"
"In a textbook. Didn't really understand."
"It summarizes properties of a probability distribution in one function. Expected value, variance, skewness, kurtosis, all derivable from it."
"Sounds convenient," Yuki said.
"Very convenient. Especially when considering sums of independent random variables."
Aoi showed a calculation. "M_X+Y(t) = M_X(t) × M_Y(t). Convolution becomes multiplication."
"Like Fourier transform," Yuki noticed.
"Exactly. Moment generating function is like Fourier transform of probability distribution."
Riku looked confused. "Everything's connected?"
"Yes. Mathematics is one large structure," Aoi said quietly. "Probability theory, information theory, Fourier analysis, all connected at the root."
Yuki looked out the window. "Maybe life is also a probability distribution."
"Philosophical," Aoi smiled. "But might be true. We're living realized values of a probability distribution."
"What's my life's expected value?" Riku asked.
"That's for you to decide," Aoi answered. "Expected value changes with choices."
"What about variance?"
"Same. Whether to take risks or choose stability."
Yuki laughed. "After school talking only about probability is surprisingly deep."
"Probability is deep," Aoi said. "Accept uncertainty, yet still do your best. That's probability theory's teaching."
The three pondered for a while about the probability distribution that is their lives.