Probability Chapter 4: Special Distributions

<p>The probability of success on a single trial. (D)</p> Signup and view all the answers

<p>It counts the number of successes. (D)</p> Signup and view all the answers

<p>Differentiate M_X(t) and evaluate at t=0. (B)</p> Signup and view all the answers

<p>Probability mass function. (A)</p> Signup and view all the answers

<p>q (A)</p> Signup and view all the answers

<p>$g(x; p) = p q^{x-1}$ (A)</p> Signup and view all the answers

<p>0.0117 (C)</p> Signup and view all the answers

<p>X has a geometric distribution parameterized by p. (C)</p> Signup and view all the answers

<p>$rac{1}{p}$ (A)</p> Signup and view all the answers

<p>Geometric distributions utilize geometric series to calculate probabilities. (C)</p> Signup and view all the answers

<p>All trials are identical and independent. (D)</p> Signup and view all the answers

<p>$rac{q}{p^2}$ (A)</p> Signup and view all the answers

<p>$G(x; p) = 1 - q^x$ (D)</p> Signup and view all the answers

<p>Hypergeometric distribution (A)</p> Signup and view all the answers

<p>Number of items drawn of type 1 (B)</p> Signup and view all the answers

<p>X ~ HYP(n, N, M) (A)</p> Signup and view all the answers

<p>h(x; n, N, M) (A)</p> Signup and view all the answers

<p>max(0, n-N+M) ≤ x ≤ min(n, M) (A)</p> Signup and view all the answers

<p>Deciding whether to accept a lot of manufactured items (D)</p> Signup and view all the answers

<p>No more than 3 defective items may be drawn (C)</p> Signup and view all the answers

<p>Total number of items of type 1 (B)</p> Signup and view all the answers

<p>Exactly two outcomes per trial. (C)</p> Signup and view all the answers

<p>The probability of failure in a single trial. (B)</p> Signup and view all the answers

<p>The probability mass function mirrors the coefficients in the binomial expansion. (B)</p> Signup and view all the answers

<p>Both n and p can vary in different applications. (D)</p> Signup and view all the answers

<p>To provide a framework for application and analysis. (D)</p> Signup and view all the answers

<p>The probabilities for all outcomes must equal 1. (D)</p> Signup and view all the answers

<p>$M_X(t) = (p e + q)^n$ (A)</p> Signup and view all the answers

<p>It is one possible outcome based on defined probability. (B)</p> Signup and view all the answers

<p>$np$ (A)</p> Signup and view all the answers

<p>$npq$ (D)</p> Signup and view all the answers

<p>p must lie between 0 and 1 inclusive. (B)</p> Signup and view all the answers

<p>It can be used to derive expectations and variances. (B)</p> Signup and view all the answers

<p>$np$ (C)</p> Signup and view all the answers

<p>$2.5$ (B)</p> Signup and view all the answers

<p>X (B)</p> Signup and view all the answers

<p>The number of successes is fixed. (D)</p> Signup and view all the answers

<p>$4$ (B)</p> Signup and view all the answers

<p>$rac{nM(N - M)(N - n)}{N^2(N - 1)}$ (D)</p> Signup and view all the answers

<p>It represents the number of events in a fixed interval of time or space. (C)</p> Signup and view all the answers

<p>The average number of events in a fixed interval. (B)</p> Signup and view all the answers

<p>When n approaches infinity, and p approaches 0 while np is constant. (D)</p> Signup and view all the answers

<p>$f(x; ext{λ}) = rac{e^{- ext{λ}} ext{λ}^x}{x!}$ for $x = 0, 1, 2,...$ (D)</p> Signup and view all the answers

<p>The expected value is equal to λ. (C)</p> Signup and view all the answers

<p>The total number of defective items in the population. (C)</p> Signup and view all the answers

Mathematical function that encapsulates the probability distribution of a random variable. It's computed by averaging e^tx over all possible values of x, where x is the random variable and t is a parameter.

The expected value (average) of a random variable.

A measure of how spread out a distribution is - the average squared deviation from the mean.

A discrete probability distribution that describes the probability of obtaining a certain number of successes in a fixed number of independent Bernoulli trials.

The probability of success in a single Bernoulli trial.

The probability of failure in a single Bernoulli trial.

The formula for the M.G.F. of a binomial distribution.

The mean and variance of a binomial distribution are directly calculated from the probability of success 'p', and the number of trials 'n'.

A Bernoulli trial is a single experiment with two possible outcomes: success or failure. Each trial is independent of the others.

The total number of trials in a binomial distribution, represented by the letter 'n'.

The probability mass function (p.m.f.) of a binomial distribution gives the probability of observing exactly x successes in n trials. It is calculated as: b(x; n, p) = (n choose x) * p^x * (1-p)^(n-x).

The binomial distribution gets its name from the binomial theorem, which expands the power of a binomial expression (q + p)^n. The terms of the binomial expansion correspond to the probabilities of different numbers of successes in the binomial distribution.

The sum of all probabilities for all possible values of x in a binomial distribution always equals 1. This ensures that the total probability of all possible outcomes is 1.

The binomial distribution is a discrete distribution, meaning that the random variable can only take on whole number values (like 0, 1, 2, etc.).

Is the number of trials needed to obtain the first success in a series of independent Bernoulli trials.

The probability of getting the first success on the Xth trial is p(1-p)^(x-1).

The expected value or mean of the geometric distribution is 1/p.

The variance of the geometric distribution is (1-p)/p^2.

A negative binomial distribution with k=1.

The number of trials required to get the kth success in a series of independent Bernoulli trials.

The probability of obtaining the kth success on the Xth trial is (X-1 choose k-1) p^k (1-p)^(X-k).

The expected value of the negative binomial distribution is k/p.

A probability distribution describing the probability of selecting a certain number of successes (items of type 1) from a finite population without replacement, given the total number of items, the number of success items and the sample size.

The possible values of the random variable X in a hypergeometric distribution, representing the number of success items in a sample.

The probability mass function (PMF) of the hypergeometric distribution, giving the probability of observing exactly x success items in a sample of size n.

A compact notation indicating that a random variable X follows the hypergeometric distribution with parameters n, N, and M, representing sample size, total population size, and number of success items respectively.

The process of drawing items from a finite population without replacement, where each item drawn is not returned to the population, affecting the probability of subsequent draws.

A sample of items drawn from a finite population without replacement.

An application of the hypergeometric distribution where the acceptance of a lot (collection) of items is based on the number of defective items found in a random sample.

A real-world example where the hypergeometric distribution is used to analyze the probability of finding a certain number of defective items in a sample of microchips.

The probability of getting the first success on the 'x'th trial in a sequence of independent Bernoulli trials, where 'p' is the probability of success on each trial.

The probability of getting the first success after a specific number of trials, calculated as the sum of the probabilities of getting the first success on each of the individual trials.

A mathematical function that captures the concentration of a probability distribution. For a geometric distribution, it expresses how the probability of success is distributed over the number of trials.

The expected value of a random variable following a geometric distribution. It represents the average number of trials required to achieve the first success.

The notation used to indicate that a random variable 'X' follows a geometric distribution with probability of success 'p'.

The probability of getting the first success after 'x' trials, represented as q raised to the power of 'x', where 'q' is the probability of failure in a single trial.

A distribution used to model the probability of a specific number of events occurring within a fixed interval of time or space, given the average rate of occurrence.

The average rate of occurrence of an event in the Poisson distribution.

The number of events that occur within a fixed interval of time or space in the Poisson distribution.

The Poisson distribution is a limiting case of the binomial distribution when the number of trials approaches infinity and the probability of success approaches zero.