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Random variables and expectation
Probability is a measure on events: non-negative, total 1, additive over disjoint events. Conditional probability P(A|B) = P(A ∩ B)/P(B), and Bayes reverses the conditioning. Picture it: a tree diagram; multiply along branches, add across leaves. Think it: the axioms are measure theory with total mass 1, which is why the two subjects merged.
Isibonelo esisebenza · 1 - (5/6)^4
Evaluate 1 - (5/6)^4
1 - \left(\frac{5}{6}\right)^{4}
Isigaba
1 - \left(\frac{5}{6}\right)^{4} = 1 - \frac{625}{1296}
Power: (5/6)^4 = 625/1296.
1 - \frac{625}{1296} = \frac{671}{1296}
Simplify.
Bonisa impendulo
\frac{671}{1296} \approx 0.51775
Manje
Khetha izinkinga, noma ubhale noma udwebe zakho. Isigaba ngasinye, isithombe, impendulo ifihlakele kuze kube ubuza.
Qhubeka usebenza
I-akhawunti emahhala ifaka iziphawulo kulesifundo ngasinye, irekhodi lokuthi yini oqedile, izinkinga zakho ezixazululiwe endaweni eyodwa, nodokotela owaziyo ongabuza mayelana nalekhasi. I-matematics ivulekile kubo bonke, noma ngabe babhalise noma cha.
Bhala
Ngena
Amaphawu asetshenziswa lapha
Chofoza noma yiluphi uphawu lwesibonakaliso ukuze uthole ukuchaza okuphelele, isithombe, kanye nokuthi iyiphi inhlamvu ekhona ekhuluma ngaso.
P(A) probability, conditional probability Chance of A; chance of A given that B happened. \approx approximately equal Equal to the precision shown, not exactly. \leq less/greater than or equal Inequalities that allow equality; < and > exclude it. n! factorial n × (n−1) × … × 1; the number of orderings of n things. 0! = 1. \binom{n}{k} binomial coefficient, "n choose k" Number of k-element subsets of n things: n!/(k!(n−k)!). \sum_{k=1}^{n} a_k summation Add a_k for k = 1 up to n. \int f(x) integral Antiderivative (indefinite) or signed area from a to b (definite). A \cup B union, intersection, difference In either; in both; in A but not B. \bar{x} sample mean, population mean Average of the data; average of the whole population. \sigma standard deviation, sample s.d., variance Typical distance from the mean; its square. E[X] expected value, variance Probability-weighted average of X; its spread. N(\mu normal distribution, z-score The bell curve with mean μ and variance σ²; (x − μ)/σ. \mu(A) measure of A Size of a set; the family of sets that can be measured.
Amaphawu kulesikhasi
Chofoza enye ukuze ubone ukuthi iyini inkinga
P(A) probability › \approx approximately equal › \leq less/greater than or equal › n! factorial › \binom{n}{k} binomial coefficient › \sum_{k=1}^{n} a_k summation › \int f(x) integral › A \cup B union › \bar{x} sample mean › \sigma standard deviation › E[X] expected value › N(\mu normal distribution › \mu(A) measure of A ›
Indlela: Sample spaces and the axioms
Power: (5/6)^4 = 625/1296. Simplify.
Imibuzo abantu bebuza
What is the difference between probability and statistics? Probability goes from a known model to what the data should look like; statistics goes from data back to the model. Probability theory is the deductive half.
What does the law of large numbers promise? That the average of many independent samples converges to the expected value. It says nothing about any single trial.
Okuningi Probability Theory
Random variables and expectation The common distributions The law of large numbers and the central limit theorem