Probability Chapter 4: Special Distributions

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Questions and Answers

What is the expected value, E[X], for the random variable X with the geometric distribution?

  • $\frac{1}{p}$ (correct)
  • $\frac{k(1 - q)}{p}$
  • $\frac{k}{p}$
  • $p$

Which formula represents the variance, var(X), of a geometric random variable?

  • $\frac{q}{p^2}$
  • $\frac{1}{p}(1 - p)$ (correct)
  • $\frac{q}{p} - \left(\frac{1}{p}\right)^2$
  • $\frac{1 - p}{p^2}$

In the negative binomial distribution, what happens when k=1?

  • It becomes a Poisson distribution.
  • It becomes a standard normal distribution.
  • It is transformed into a uniform distribution.
  • It is called the geometric distribution. (correct)

What does the parameter p represent in the context of the geometric distribution?

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

Which of the following is NOT a characteristic of the geometric distribution?

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

How is the generating function M_X(t) differentiated to find E[X]?

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

What does the term p.m.f. stand for in the context of a geometric random variable?

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

For the geometric distribution, which value indicates the probability of 'failure' in a given trial?

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

What is the geometric distribution formula for the probability mass function given in the content?

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

If $p = 0.75$, what is the probability that the first success occurs on the fourth trial?

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

What does the notation $X ~ GEOM(p)$ signify?

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

What is the mean (μ) of the geometric distribution as provided in the content?

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

What is the relationship between the geometric series and the geometric distribution mentioned in the content?

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

In the context of the geometric distribution, what can be inferred about the trials?

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

What is the variance ($ ext{σ}^2$) of the geometric distribution?

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

Which of the following represents the cumulative distribution function (CDF) for the geometric distribution?

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

What type of distribution is used when sampling without replacement from a finite collection of items?

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

In a hypergeometric distribution, which of the following defines the random variable X in the context of drawing items?

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

When is the hypergeometric random variable X defined in terms of M, N, and n?

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

What is the formula for the probability mass function (p.m.f.) of a hypergeometric distribution?

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

In the given context of hypergeometric distribution, which condition must be satisfied for x?

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

Which application is particularly relevant for utilizing the hypergeometric distribution in industry?

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

If 10 microchips are drawn without replacement from a box containing 100 microchips (80 good and 20 defective), what is the condition for the lot to be considered acceptable?

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

What does the letter 'M' represent in the hypergeometric distribution?

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

Which conditions are required for a random variable to be considered as having a binomial distribution?

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

In the binomial distribution, what does the parameter 'q' represent?

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

What is the relationship between the binomial distribution and the binomial expansion?

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

Which of the following statements is true regarding the parameters 'n' and 'p' in a binomial distribution?

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

What is the primary purpose of defining the parameters in a probability distribution?

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

What does the summation formula for the binomial distribution indicate about the probabilities?

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

What is the formula for the moment generating function of the binomial distribution?

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

In a series of Bernoulli trials, what characterizes a 'success'?

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

Which of the following represents the mean of the binomial distribution?

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

What does the variance of the binomial distribution equal?

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

What is the restriction on the parameter 'p' in a binomial distribution?

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

Which of the following correctly describes the use of the moment generating function?

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

When differentiating the moment generating function $M_X(t)$ with respect to $t$ to find the mean, what result is obtained when $t = 0$?

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

If a binomial distribution has parameters $n = 10$ and $p = 0.5$, what is its variance?

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

For a negative binomial distribution seeking the $k^{th}$ success, what variable represents the number of trials required?

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

In the context of the negative binomial distribution, which of the following is true?

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

What is the mean of the hypergeometric distribution given the parameters n=10, N=100, and M=20?

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

What is the variance formula for the hypergeometric distribution?

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

Which of the following correctly describes the Poisson distribution?

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

What does the parameter λ represent in the Poisson distribution?

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

For which conditions does the Poisson distribution serve as a limiting form of the binomial distribution?

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

What is the probability mass function (p.m.f.) for the Poisson distribution?

<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

If X follows the Poisson distribution with parameter λ, which of the following is true about the expected value?

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

In the context of the hypergeometric distribution, what does the parameter M signify?

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

Flashcards

Moment Generating Function (M.G.F.)

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.

Mean (μ)

The expected value (average) of a random variable.

Variance (σ²)

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

Binomial Distribution

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

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Probability of Success (p)

The probability of success in a single Bernoulli trial.

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Probability of Failure (q)

The probability of failure in a single Bernoulli trial.

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M.G.F. of Binomial Distribution

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

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Mean and Variance of 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'.

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Bernoulli Trial

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

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Number of Trials (n)

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

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Binomial Probability Mass Function (p.m.f.)

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).

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Binomial Theorem Connection

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.

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Sum of Probabilities in 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.

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Discrete Nature of Binomial Distribution

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

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Geometric Distribution

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

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Geometric Distribution Probability

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

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Expected Value of Geometric Distribution

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

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Variance of Geometric Distribution

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

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Geometric Distribution as a Special Case

A negative binomial distribution with k=1.

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Negative Binomial Distribution

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

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Probability of Negative Binomial Distribution

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

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Expected Value of Negative Binomial Distribution

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

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Hypergeometric Distribution

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.

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Possible Values of X in Hypergeometric Distribution

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

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Hypergeometric PMF

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

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X ~ HYP(n, N, M)

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.

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Sampling Without Replacement

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.

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Sample Without Replacement

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

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Acceptance Sampling with Hypergeometric Distribution

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.

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Microchip Acceptance Example

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.

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Geometric Probability Mass Function (PMF)

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.

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Geometric Cumulative Distribution Function (CDF)

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.

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Geometric Moment Generating Function (MGF)

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.

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Mean of Geometric Distribution

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

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X ~ GEOM(p)

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

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Probability of exceeding 'x' trials (P(X > x))

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.

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Poisson Distribution

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.

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λ (Lambda)

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

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X (Random Variable)

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

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Poisson as a Limit of Binomial

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.

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Study Notes

Chapter 4: Special Probability Distributions

  • This chapter studies common probability distributions, examining their properties for theoretical and practical applications.
  • It begins with discrete distributions, followed by continuous distributions.

4.1 Introduction

  • Chapters 1 and 2 introduced general probability distributions.
  • This chapter focuses on specific probability distributions, analyzing their core characteristics.
  • The chapter's findings are crucial for diverse applications.

4.2 Discrete Distributions

  • This section explores common discrete probability distributions.
  • It describes the key characteristics of these distributions.

I- The Binomial Distribution

  • A series of independent Bernoulli trials are examined where each trial results in either "success" or "failure".

  • Probability (p) of success remains consistent across all trials.

  • The random variable (r.v.) X represents the total number of successes in 'n' trials.

  • X follows a binomial distribution.

  • Definition 4.1: The random variable X follows a binomial distribution with parameters n and p if its probability mass function (PMF) is given by:

    b(x; n, p) = f(x) = ( n! / (x! * (n-x)!)) * p^x *(1-p)^(n-x)

    for x = 0, 1, 2, ..., n where 0 ≤ p ≤ 1 and q = 1-p.

  • The parameters n and p represent constants unique to each distribution application.

Example 4.1

  • Describes an example of a company producing computer chips where the probability of a defective chip is 40%.
  • Seven chips are randomly selected, determining the probability of particular scenarios (e.g., exactly 3 defective chips, 5 or more defective chips, etc.)

Example 4.2

  • Demonstrates a fair die rolling scenario.
  • Determines number of rolls necessary to achieve a probability of at least 0.5 for obtaining at least one 6.

II- The Negative Binomial Distribution

  • This section investigates a sequence of independent Bernoulli trials, focusing on the number of trials needed to achieve a specified number of successes.

  • The random variable X represents the number of trials required for a kth success.

  • Definition 4.2: The random variable X follows a negative binomial distribution with parameters k and p if its probability mass function (PMF) is given by:

    b (x; k, p) = ( x-1; k-1) * p^ k *q^(x-k)

    for x = k, k+1, k+2, ...

    where 0 ≤ p ≤ 1 and q = 1-p.

Example 4.3

  • A child's probability of catching a contagious disease is 0.4.
  • Probability that the 10th child exposed to the disease will be the third to catch it is calculated.

Example 4.4

  • Team A and Team B participate in a seven-game series.
  • Probability that the series ends in exactly six games is calculated.

V- The Poisson Distribution

  • The random variable (r.v.) X follows a Poisson distribution with parameter λ > 0 if its probability mass function (PMF) is given by:

    f(x; λ) = (e^−λ * λ^x) / x! for x = 0, 1, 2, ...

Example 4.7

  • Calculates the probability of obtaining 3 defective transistors when 1% of transistors are defective, and 100 transistors are selected.

Example 4.8

  • Calculates probabilities for defective bindings in 400 books when 2% have defective bindings.

4.3 Continuous Distributions

  • This section analyses well-known continuous probability distributions.
  • Various properties of these distributions are described.

I- The Uniform (Rectangular) Distribution

  • A continuous random variable (r.v) X can take on values within a bounded interval (α, β)

  • The probability density function (PDF) is constant within this interval (α < x < β)

  • This distribution is uniform within the given interval (α, β)

  • Definition 4.6: The continuous r.v. X has a uniform distribution with parameters a and β if its PDF is:

    f(x) = 1/(β - α) for α < x < β. f(x)= 0 otherwise.

II- The Exponential Distribution

  • Definition 4.7: The continuous r.v X has exponential distribution with parameter 0>0 if its PDF is:

    f(x, 0) = (1/θ) *e^(-x/θ) for x > 0. f(x) = 0 otherwise

  • Theorem 4.7: Shows the exponential distribution exhibits the memoryless property.

III- The Normal Distribution

  • Definition 4.8: The continuous r.v. X has a normal distribution with parameter μ and σ if its PDF is:

    f(x, μ, σ) = (1/(σ√2π)) *e^(-(x-μ)²/2σ²) for -∞ < x < ∞

  • Theorem 4.9: Provides the moment generating function (MGF) for the normal distribution.

  • Examples demonstrate the use of the normal distribution in probability calculations, calculating probabilities, and solving related problem scenarios involving mean, standard deviation, and various areas related to the distribution curve.

Various Exercises

  • The section contains various exercises illustrating typical applications of these probability distributions.

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