2.4 Statistical Distributions

Cards (84)

  • A statistical distribution describes the possible values and their associated probabilities
  • The probability of each value in a discrete probability distribution is given by the probability mass function
  • The Poisson distribution uses a single parameter called the rate parameter
  • The binomial distribution is a discrete distribution that analyzes successes in repeated trials.
    True
  • The Poisson distribution is used to model the number of events in a fixed interval of time
  • The Binomial distribution has two parameters: the number of trials and the success probability
  • Match the distribution with its use case:
    Binomial ↔️ Independent trials with two outcomes
    Poisson ↔️ Counting events in fixed intervals
  • Three common statistical distributions are the normal, binomial, and Poisson
  • Statistical distributions allow us to understand the underlying patterns of a dataset.

    True
  • What are the parameters of the normal distribution?
    Mean and standard deviation
  • What is one purpose of statistical distributions in data analysis?
    Understand dataset patterns
  • The Poisson distribution models the number of events in a fixed period
  • The Exponential distribution is used to analyze the time intervals between events.

    True
  • The Poisson distribution models events occurring in a fixed time or space interval.
    True
  • What is a statistical distribution?
    Mathematical model of values
  • Match the distribution with its key use case:
    Normal ↔️ Modeling continuous data
    Binomial ↔️ Analyzing successes in trials
    Poisson ↔️ Modeling events in a fixed period
  • The probability of an event in a continuous distribution is determined by integrating the probability density function
  • Match the distribution with its parameter:
    Normal ↔️ Mean and standard deviation
    Binomial ↔️ Number of trials and success probability
    Poisson ↔️ Rate parameter
  • The Binomial Distribution has two parameters: the number of trials \(n\) and the success probability p
  • What is the parameter of the Poisson Distribution?
    \(\lambda\) (rate parameter)
  • The PDF of the Normal Distribution is \( f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \), where \(\mu\) represents the mean
  • Match the distribution with its parameters:
    Normal ↔️ Mean (\(\mu\)), Standard deviation (\(\sigma\))
    Exponential ↔️ \(\lambda\) (rate parameter)
  • Continuous probability distributions assign probabilities to a continuous range of values
  • What is the Normal distribution used for?
    Modeling continuous data
  • What is the Exponential distribution used for?
    Analyzing time intervals
  • Some common types of statistical distributions include the normal, binomial, and Poisson
  • What is the parameter of the Poisson distribution?
    Rate parameter
  • Discrete probability distributions use the Probability Mass Function (PMF) to define probabilities
  • The Poisson Distribution uses the rate parameter \(\lambda\) to describe the average rate of events
  • What is the parameter of the Poisson Distribution?
    Rate parameter (\(\lambda\))
  • Match the distribution with its PDF:
    Normal ↔️ f(x)=f(x) =1σ2πe−(x−μ)22σ2 \frac{1}{\sigma\sqrt{2\pi}} e^{ - \frac{(x - \mu)^{2}}{2\sigma^{2}}}
    Exponential ↔️ f(x)=f(x) =λe−λx \lambda e^{ - \lambda x}
  • The Normal Distribution models symmetrical data and has parameters for mean and standard
  • The exponential distribution is used to analyze time intervals between events.
  • The Poisson distribution uses the parameter λ to describe the average event rate.
  • Match the distribution with its key application:
    Normal ↔️ Modeling continuous data
    Binomial ↔️ Analyzing success rates
    Poisson ↔️ Estimating event numbers
    Exponential ↔️ Studying time intervals
  • Showing your work in exams can earn partial credit even if you don't get the correct answer.
  • A probability distribution describes the likelihood of different values of a random variable.

    True
  • The binomial distribution is used to analyze the number of successes in a series of independent trials.

    True
  • The normal distribution is used to model continuous data such as heights or weights
  • What is the formula for the probability mass function (PMF) of the binomial distribution?
    P(X=k)=P(X = k) =(nk)pk(1−p)n−k \binom{n}{k} p^{k} (1 - p)^{n - k}