The exponential distribution models the wait to the next event in a constant-rate Poisson process. Write X ~ Exp(λ) with rate λ > 0. If time is measured in years, λ has units of events per year and X has units of years. Larger rates mean shorter typical waits...
A continuous uniform variable assigns equal probability to equal-length subintervals inside a fixed range. Write X ~ Uniform(a,b) with a < b. The density is constant on [a,b], but no individual point has positive probability...
The normal distribution is a symmetric continuous model with a center and a scale. Write X ~ N(μ,σ²), where μ is the mean and σ > 0 is the standard deviation. Increasing μ shifts the distribution; increasing σ spreads its probability over a wider range...
The gamma family models positive quantities with flexible right-skewed shapes. Write X ~ Gamma(α,β), where α > 0 is the shape and β > 0 is the rate. This note uses a rate, not a scale; the corresponding scale is θ = 1/β...
Use these expandable questions to test explanations, not just terminology. Before opening an answer, predict the result and try a small browser example. The categories cover HTML, CSS, JavaScript, protocols and hosting. Examples are illustrative; versions, browser support and hosting prices should b...
Program może potrzebować losowego wyniku, na przykład podczas rzutu kostką, tasowania kart albo symulacji. Komputer zwykle nie losuje jednak każdej liczby od zera. W C++ najczęściej korzysta z generatora pseudolosowego: algorytmu, który tworzy długą, pozornie losową sekwencję wartości...
Program często pobiera tekst, przechowuje go, porównuje i wyświetla. Zanim wybierzesz funkcję do pracy z tekstem, trzeba rozróżnić trzy rzeczy: znak (np. A), jego zapis w pamięci (jeden lub kilka bajtów) oraz napis (ciąg znaków w określonej reprezentacji). To rozróżnienie wyjaśnia, dlaczego długość ...
Poprzednia notatka pokazała, że tekst i inne dane są przechowywane w pamięci jako bajty. Bajt składa się z bitów — najmniejszych jednostek, które mają wartość 0 albo 1. Operacje bitowe pozwalają sprawdzić lub zmienić wybrane bity liczby. Najczęściej używa się ich do flag, masek, pól protokołów i dan...
W poprzednich notatkach tekst był ciągiem bajtów, a maska pozwalała wybrać bity opisujące stan. W programie można te dane połączyć: na przykład wiadomość ma treść i status „przeczytana”. Klasa pozwala przechowywać taki stan razem z operacjami, które go zmieniają. To podstawowa idea programowania obi...
One of the fundamental skills is to navigate and manage files and directories effectively. Here, we focus on the crucial concepts that will facilitate your work within the file system...
Try answering each question before opening its answer. SQL exercises identify their dialect; syntax and guarantees vary between engines...
Stationarity is a statement about a process across possible realizations, not a demand that one plotted path look flat. Imagine moving an observation window from one date to another. If the relevant probability laws stay the same after that common shift, information from one part of the record can d...
A backtest recreates forecasting decisions at historical dates. At each date, hide the later observations, run the procedure using the information then available, and score the resulting predictions when their target observations become available. This evaluates a forecasting process rather than how...
An autoregressive model predicts the next observation from observed values of the same series. If high demand tends to be followed by high demand, lagged demand contains predictive information. An AR equation assigns coefficients to that information and adds a new shock for the part the past did not...
A time plot tells us when values rise and fall. Frequency-domain analysis asks which oscillation rates contribute to those movements. A repeating annual pattern in monthly data appears near 1/12 cycles per month; shorter and longer patterns appear at different frequencies. This view can separate ove...
Imagine observing daily demand over the same month under many possible sequences of customer arrivals. Each possible sequence is a path. At a fixed date, the different possible demands form a random variable. A stochastic process specifies how those random variables fit together across dates, includ...
Suppose electricity demand depends on temperature, but errors from a temperature-only regression stay positive for several days in a row. The predictor explains one source of movement; the residual runs suggest another. Dynamic regression represents both the predictor relationship and the remaining ...
An MA(1) equation tells us how a current shock and a previous shock combine to form an observation. In practice, however, we observe the combination, not the shocks. To estimate residuals and forecast recursively, we need to work backward from observations to a plausible shock history. Invertibility...
Related series can carry predictive information for one another. A demand series may respond to earlier prices, while prices respond to earlier demand. Modeling each series separately ignores those cross-lag relationships. A multivariate time-series model writes equations for the variables jointly, ...
A random walk says that the current level is the previous level plus a new increment. If today's shock raises the level by 2, tomorrow starts from that raised level. There is no mean-reverting term that gradually pulls the effect back toward a fixed baseline. Each shock therefore appears once in the...
Suppose a quantity retains 70% of its previous value and receives an addition of 2 at every step. The rule determines tomorrow's value from today's value; an initial value determines which particular path follows that rule. A difference equation formalizes this kind of discrete-time evolution...
A price level, its return, and the uncertainty of that return answer different questions. The price records a cumulative level. A return measures proportional change over an interval. A volatility model describes how large an unexpected return is likely to be, even when its sign remains hard to pred...
A noisy measurement need not equal the underlying quantity we want to track. Daily observations can jump because the true level changed, because measurement error changed, or both. A state-space model writes one equation for the hidden state and another for the observation, making those sources of v...
Take two observations from the same series, subtract the appropriate mean from each, and multiply the deviations. A positive product means that both observations are on the same side of their means; a negative product means that they are on opposite sides. Autocovariance is the expectation of this p...
If today's value is high, does that help predict tomorrow's value? What about the value two periods later? The autocorrelation function (ACF) measures the overall linear association at each separation. The partial autocorrelation function (PACF) asks what remains at a separation after the intermedia...
ARMA, ARIMA, and SARIMA describe related mechanisms at increasing levels of structure. An ARMA model combines feedback from past observations with the effects of current and past shocks around a stable mean. ARIMA first takes differences to remove an integrated level, then models the changes with AR...
A series can rise over several years while repeatedly dipping at the same time each year. The rise and the repeating dip describe different structure. Trend summarizes persistent movement in level or slope; seasonality describes a pattern associated with a fixed known period. A cycle with an evolvin...
A regression can explain the response's relationship with observed predictors while leaving persistent departures around that relationship. Regression with ARMA errors models those departures as a time series. Its two equations distinguish the predictor contribution from the dynamic error; the final...
A forecast answers a dated question: what can be said about a specified future quantity using the information available now? Predicting next month's total sales from today's data is different from predicting tomorrow's sales, and both differ from estimating a past trend with the full record availabl...
A moving-average model explains an observation through a short history of shocks. A temporary disruption may affect today's measurement and leave a smaller effect tomorrow, then disappear. An MA(1) encodes that pattern with one current shock and one lagged shock; an MA(q) extends the direct response...
A time series records observations in an order that matters. A week of unusually high demand followed by a week of low demand tells us something different from the same values alternating each day. An unordered sample preserves the values but loses that pattern, even though the pattern may determine...
An AR equation predicts the next value from earlier values. The Yule-Walker equations describe what that prediction rule implies for covariance across lags. Multiply the model by an earlier centered observation, take expectations, and use the innovation's lack of covariance with the past. The result...
The symbol B means “take the previous observation.” It acts on a sequence: BXₜ = Xₜ₋₁, and applying it twice gives B²Xₜ = Xₜ₋₂. It is useful because a long equation involving many lags can then be written as a polynomial acting on the whole sequence...
A time-series model often says that today's observation contains the effects of many earlier shocks. Writing those effects as an infinite sum is convenient, but it raises a concrete question: does including more history settle on a definite answer, or does the approximation keep changing without lim...
Fitting a time-series model means more than finding coefficients for an equation. The procedure chooses a target, decides how trend and seasonality enter, handles missing observations, estimates dynamic parameters, and produces forecasts from a defined information set. Any of those decisions can aff...