آمار و احتمالProbability & Statistics
شمارش، متغیرهای تصادفی، توزیعها و استنباط آماری.
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کارتها · 16
CV = sigma/mu normalizes spread so data sets with different means can be compared for relative variability.
Split the conditioning event into disjoint pieces to expand its probability, then apply the conditional-probability definition.
Count the union of two sets by adding their sizes and subtracting the double-counted overlap.
When one hit strikes exactly one of n items uniformly, any k specific items all survive with probability (n-k)/n.
A robust, quartile-only measure of asymmetry: Skew_Q=(Q_3+Q_1-2Q_2)/(Q_3-Q_1) , always in [-1,1] , with sign set by which half of the IQR the median sits in.
Series blocks multiply reliabilities; parallel blocks multiply unreliabilities. Collapse the network inside-out, treating each block as one component.
When several groups are pooled, the total sum of squared deviations from the grand mean splits cleanly into a within-group part plus a between-group part — the discrete parallel-axis (ANOVA) identity.
Sampling k items without replacement from a population of N with K special ones: count the favorable draws sequentially, or use the hypergeometric formula — both give the same answer.
A stem-and-leaf plot lists every data value as stem + leaf; rebuild the values, sum them, and divide by the count to get the mean.
For independent events, attack 'at least one' through its complement 'none' (a product of failure probabilities), and build 'exactly one' by swapping a single factor — then collapse the product by telescoping.
Changing one data value shifts the mean by (change)/n, but leaves the median untouched as long as the order of the middle values is unchanged.
Average speed is total distance over total time, never the plain average of the leg speeds; when you know the distance fraction of each leg it equals the distance-weighted harmonic mean of the speeds.
When two distinct objects are drawn at random from a set, every unordered pair is equally likely, so the probability of any event is (favorable pairs) / C(n,2) — reducing the problem to two separate counts.
If one player tosses a fair coin one more time than another, the probability that the first gets strictly more heads is exactly tfrac12 — proved by a clean symmetry split, no binomial sums needed.
Treat objects that must stay together as one glued block, then multiply by the internal orderings — the standard tool for computing probabilities of adjacency in a random row arrangement.
A dataset has zero variance (equivalently zero standard deviation) if and only if every value is identical; then the mean, median, every percentile, and the single distinct value all collapse to that same constant.