Probability I: Distribution Functions
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Questions and Answers

What is the probability distribution function denoted as in the context of a random variable X?

  • F(x) (correct)
  • P(X ≤ x)
  • p(X)
  • f(x)

    • Which of the following properties is true for a probability distribution function F?

  • F can be negative.
  • F cannot exceed 1. (correct)
  • F is always equal to 0.
  • F decreases as x increases.

    • What does a probability mass function (PMF) describe?

  • The probability that a discrete random variable takes on any value.
  • The probability distribution of a discrete random variable. (correct)
  • The density of a continuous random variable at a point.
  • The probability distribution of a continuous random variable.

    • In a Bernoulli trial, what outcome corresponds to X=0?

    <p>Failure</p> Signup and view all the answers

    What is true about the limits of the cumulative distribution function F?

    <p>lim F(b) = 1 as b → ∞</p> Signup and view all the answers

    What type of random variable does the probability density function (PDF) describe?

    <p>Continuous random variable</p> Signup and view all the answers

    Which of the following distributions is NOT commonly associated with a discrete random variable?

    <p>Normal distribution</p> Signup and view all the answers

    What does a Bernoulli random variable assume?

    <p>Only two values</p> Signup and view all the answers

    What is the expected value of a Bernoulli random variable X?

    <p>p</p> Signup and view all the answers

    What does the variance of a Bernoulli random variable X equal?

    <p>p(1 - p)</p> Signup and view all the answers

    What are the possible values of the random variable Sn in a Binomial distribution?

    <p>0, 1, 2, ..., n</p> Signup and view all the answers

    Which of the following conditions is NOT necessary for a Binomial distribution?

    <p>The outcomes can be categorized more than two ways</p> Signup and view all the answers

    What is the probability density function for a Bernoulli random variable f(x)?

    <p>1 - p for x = 0, p for x = 1</p> Signup and view all the answers

    Which of the following is true about the sum Sn of Bernoulli trials?

    <p>It counts the number of successes in n trials</p> Signup and view all the answers

    What is the form of the pdf of a Binomial random variable X?

    <p>nCx p^x(1 - p)^n</p> Signup and view all the answers

    Which of the following values can p take in the context of Bernoulli random variables?

    <p>0 ≤ p ≤ 1</p> Signup and view all the answers

    What is the expected value of a Poisson random variable X?

    <p>$ u$</p> Signup and view all the answers

    In a Poisson distribution, what does the parameter λ represent?

    <p>The rate of occurrence over a fixed interval</p> Signup and view all the answers

    Which of the following statements about variance and standard deviation in a Poisson distribution is true?

    <p>Variance and standard deviation are equal to each other</p> Signup and view all the answers

    If a rare disease occurs in 2 percent of a population, what is the value of λ for a sample of 10,000 people?

    <p>200</p> Signup and view all the answers

    In the formula $ ext{P}(X eq 5)$, which time-sensitive event is described?

    <p>The probability of the random variable not equaling 5.</p> Signup and view all the answers

    Which exploratory data analysis technique is effective in identifying outliers?

    <p>Box plots</p> Signup and view all the answers

    When approximating a binomial distribution with a Poisson distribution, which condition must be true?

    <p>$n$ must be large and $p$ must be small.</p> Signup and view all the answers

    What feature of data does exploratory data analysis (EDA) primarily focus on?

    <p>Understanding the underlying structure, patterns, and relationships</p> Signup and view all the answers

    What is the mean or expected value of a binomial random variable X with parameters n and p?

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

    In a binomial distribution, which of the following conditions must be satisfied?

    <p>There must be a fixed number of trials.</p> Signup and view all the answers

    If a soldier has a probability of hitting a target of 0.8 and fires 10 shots, what is the variance of the number of hits?

    <p>$10(0.8)(0.2)$</p> Signup and view all the answers

    Which formula represents the probability density function for a binomial random variable X?

    <p>$P(X = x) = nCx p^x (1-p)^{n-x}$</p> Signup and view all the answers

    What is the probability that a soldier hits a target at least 9 times out of 10 shots, with a hit probability of 0.8?

    <p>$0.3758$</p> Signup and view all the answers

    For a random variable with a hypergeometric distribution, which of the following statements is true?

    <p>The population size is fixed and known.</p> Signup and view all the answers

    In the context of hypergeometric distribution, what does the term 'k' represent?

    <p>The number of successes in the population.</p> Signup and view all the answers

    What is the probability of obtaining 2 or fewer hearts when selecting 5 cards from a standard deck?

    <p>Use the hypergeometric probability formula.</p> Signup and view all the answers

    What type of experiment is described when randomly selecting 5 cards from a deck and counting the number of hearts?

    <p>Hypergeometric experiment</p> Signup and view all the answers

    Which formula is used to calculate the probability of a hypergeometric random variable?

    <p>$ rac{k!}{x!(k-x)!} rac{(N-k)!}{(n-x)!(N-n)!}$</p> Signup and view all the answers

    What is the expected value formula for a hypergeometric random variable?

    <p>$ rac{nk}{N}$</p> Signup and view all the answers

    What does the variable $ ext{λ}$ represent in the context of the Poisson random variable?

    <p>The rate of occurrences in a Poisson distribution</p> Signup and view all the answers

    Which of the following formulas correctly defines the probability density function of a Poisson variable?

    <p>$ rac{λ^x e^{-λ}}{x!}$</p> Signup and view all the answers

    In a hypergeometric distribution, what is the relationship between the variance and the number of trials?

    <p>Variance decreases as the number of trials increases</p> Signup and view all the answers

    Which condition qualifies an event as a rare event in a Poisson experiment?

    <p>The probability of success is very small</p> Signup and view all the answers

    What is the standard deviation formula for a hypergeometric random variable?

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

    Study Notes

    Course Information

    • Course Title: Probability I
    • Course Code: STA 112
    • University: Bowen University
    • Location: Iwo, Nigeria
    • College: College of Agriculture, Engineering and Sciences
    • Credits: 3
    • Instructor: Daniel Akinboro
    • Program: Statistics Programme

    Probability Distribution Function

    • Let X be a random variable with probability density function f(x).
    • The probability distribution function, F(x), is defined as Σf(y) for x real, where the summation is over all y ≤ x.
    • F(x) = P(X ≤ x)
    • Properties of the probability distribution function (PDF):
      • F is a non-decreasing function. If a < b, then F(a) < F(b).
      • Limit as b approaches ∞ of F(b) = 1.
      • Limit as b approaches -∞ of F(b) = 0.
      • F is right-continuous. This means F(b +) = F(b).

    Probability Mass Function (PMF) and Probability Density Function (PDF)

    • PMF: Used to describe the probability distribution of a discrete random variable.
    • PMF gives the probability that a discrete random variable takes on a specific value.
    • PDF: Used to describe the probability distribution of a continuous random variable.
    • PDF gives the probability density of a continuous random variable at a specific point.

    Probability Distributions

    • Bernoulli Distribution: Describes a random variable with only two possible outcomes (success or failure).
    • Binomial Distribution: Sum of independent Bernoulli random variables.
      • Describes the probability of a specific number of successes in a fixed number of trials.
    • Hypergeometric Distribution: Describes the probability of a specific number of successes in a fixed number of draws without replacement from a finite population.
    • Poisson Distribution: Describes the probability of a specific number of events in a fixed interval of time or space.
    • Normal Distribution: A continuous probability distribution. Has a characteristic bell shape. Specified by mean and variance.

    Bernoulli Random Variables (cont.)

    • Probability Density Function (PDF) of X is:
      • P(X = 1) = p, P(X = 0) = 1 - р
      • f(x) = px(1 - p)1-x, x = 0, 1 (0 ≤ p ≤ 1)
      • f(x) = 0 elsewhere
    • Expectation, Variance and Standard Deviation of a Bernoulli random variable X:
      • E(X) = p
      • Var(X) = p(1 – p)
      • SD(X) = √p(1 - p)

    Binomial Random Variable

    • Describes the number of successes in n independent Bernoulli trials.
    • Probability of success in each trial is p.
    • Possible values of the variable range from 0 to n.
    • Parameters: n (number of trials) and p (probability of success).

    Binomial Random Variable (cont.)

    • Definition: A discrete random variable X, denoting the total number of successes in n trails, is said to have the binomial distribution if :
      • P(X = x) = nCx * px * (1-p)n-x where x = 0, 1, ..., n, and 0 ≤ p ≤ 1
    • Conditions for a Binomial Distribution:
      • Fixed number of trials
      • Only two possible outcomes (success or failure)
      • Independent trials
      • Constant probability of success

    Binomial Random Variable (cont.)

    • Expectation and Variance of a Binomial Random Variable X:
    • E(X) = np
    • Var(X) = np(1 - p)
    • SD(X) = √np(1 - p)

    Hypergeometric Random Variable

    • Describes the probability of drawing exactly x successes from a population of N items, where k of these are successes, and n items are drawn.
    • Formula: P(X = x; N, n, k) = kCx * (N – k)Cn – x/ NCn

    Hypergeometric Random Variable (cont.)

    • Expectation and Variance of a Hypergeometric random variable X:
    • E(X) = nk/N
    • Var(X) = nk(N – k)(N – n)/N²(N – 1)
    • SD(X) = √nk(N – k)(N – n)/N²(N – 1)

    Poisson Random Variable

    • Describes the probability of a specific number of events in a given interval, when the events are rare.
    • The probability of an event happening is very small, while the number of trials is large.
    • Formula: P(X = x) = (λ^x * e^-λ)/x! , where x = 0, 1, 2,...

    Poisson Random Variable (cont.)

    • Example of a rare event:
      • Rate of accidents per month
    • Calculating parameters
    • Using the rate of the occurrence of events or successes over a specific time interval
      • Mean or average rate (λ)
      • Variance and standard deviation is equal to λ
      • Probability distribution of occurrence of events.

    Normal Random Variable

    • A continuous random variable with a characteristic bell-shaped distribution.
    • Defined by its mean (μ) and variance (σ²).
    • Probability Density Function (PDF):
      • f(x; μ, σ²) =1/√(2 π σ²) * e ^(-(x-μ)² / (2σ²))

    Normal Random Variable (cont.)

    • Standardized Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.

    Exploratory Data Analysis (EDA)

    • Involves analyzing and visualizing data to understand its underlying structure, patterns and relationships.
    • Techniques include histograms, scatter plots, and box plots.
    • Helps identify outliers, trends and potential variables.

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    Probability I Past Paper PDF

    Description

    This quiz covers the essential concepts of probability distribution functions (PDF) and probability mass functions (PMF) in the context of Probability I. Students will explore the properties and definitions, as well as the implications of these functions for random variables. Prepare to test your understanding of key principles in probability theory.

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