Probability Rules! (7.1)
... If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. P(A) + P(B) = P(C) Example: Tossing a coin three times... Let event A = getting 2 heads and 1 tail Let event B = getting 3 heads What's the probability of getting more ...
... If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. P(A) + P(B) = P(C) Example: Tossing a coin three times... Let event A = getting 2 heads and 1 tail Let event B = getting 3 heads What's the probability of getting more ...
ST 371 (IV): Discrete Random Variables
... Remarks: So far, we have been defining probability functions in terms of the elementary outcomes making up an experiment’s sample space. Thus, if two fair dice were tossed, a probability was assigned to each of the 36 possible pairs of upturned faces. We have seen that in certain situations some att ...
... Remarks: So far, we have been defining probability functions in terms of the elementary outcomes making up an experiment’s sample space. Thus, if two fair dice were tossed, a probability was assigned to each of the 36 possible pairs of upturned faces. We have seen that in certain situations some att ...
Review for Midterm II
... 2. How to calculate probabilities, means, and variances for each type of discrete random variable we've studied. 3. How to use a moment-generating function to find a mean and variance or to identify a p.m.f of a r.v. X 4. Application of the Poisson distribution, and the Poisson approximation to the ...
... 2. How to calculate probabilities, means, and variances for each type of discrete random variable we've studied. 3. How to use a moment-generating function to find a mean and variance or to identify a p.m.f of a r.v. X 4. Application of the Poisson distribution, and the Poisson approximation to the ...
Probability and statistics
... a) two of them want to sit together, b) two of them want to sit together on an edge of the bench? ...
... a) two of them want to sit together, b) two of them want to sit together on an edge of the bench? ...
08-nonmeasurable set
... countably infinite number of equally probably events. • Assuming Axiom of Choice, there exists a nonmeasurable set. • Do you know how to show there is a subset S in a 1 x 1 square such that the area of S is undefined? ...
... countably infinite number of equally probably events. • Assuming Axiom of Choice, there exists a nonmeasurable set. • Do you know how to show there is a subset S in a 1 x 1 square such that the area of S is undefined? ...
Probability spaces • Discrete random variables - E
... Ω = {ω i , i = 1, . . . , n} for some integer n, and countable if Ω = {ω i , i = 1, 2, . . .}; otherwise it is uncountable. Example: The sample space of the previous example is Ω = {ω 1 , ω 2 , ω 3 , ω 4 } = {HH, HT, TH, TT}, where, e.g., HT means “first toss heads (H), second toss tails (T)”. ...
... Ω = {ω i , i = 1, . . . , n} for some integer n, and countable if Ω = {ω i , i = 1, 2, . . .}; otherwise it is uncountable. Example: The sample space of the previous example is Ω = {ω 1 , ω 2 , ω 3 , ω 4 } = {HH, HT, TH, TT}, where, e.g., HT means “first toss heads (H), second toss tails (T)”. ...
Randomness
Randomness is the lack of pattern or predictability in events. A random sequence of events, symbols or steps has no order and does not follow an intelligible pattern or combination. Individual random events are by definition unpredictable, but in many cases the frequency of different outcomes over a large number of events (or ""trials"") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will occur twice as often as 4. In this view, randomness is a measure of uncertainty of an outcome, rather than haphazardness, and applies to concepts of chance, probability, and information entropy.The fields of mathematics, probability, and statistics use formal definitions of randomness. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.Randomness is most often used in statistics to signify well-defined statistical properties. Monte Carlo methods, which rely on random input (such as from random number generators or pseudorandom number generators), are important techniques in science, as, for instance, in computational science. By analogy, quasi-Monte Carlo methods use quasirandom number generators.Random selection is a method of selecting items (often called units) from a population where the probability of choosing a specific item is the proportion of those items in the population. For example, with a bowl containing just 10 red marbles and 90 blue marbles, a random selection mechanism would choose a red marble with probability 1/10. Note that a random selection mechanism that selected 10 marbles from this bowl would not necessarily result in 1 red and 9 blue. In situations where a population consists of items that are distinguishable, a random selection mechanism requires equal probabilities for any item to be chosen. That is, if the selection process is such that each member of a population, of say research subjects, has the same probability of being chosen then we can say the selection process is random.