Discrete Uniform Distribution: Mean, Variance, Moment Generating Functions
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Questions and Answers

For a uniform distribution U(1,6), the probability mass function (PMF) is defined as P(X=x) = ?

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  • 1/6 (correct)

    • What is the variance of a uniform distribution U(1,6)?

  • 7/12 (correct)
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    • If a uniform distribution U(1,6) has a moment generating function MX(t), what would be the value of MX(0)?

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  • 1 (correct)

    • What is the mean of a uniform distribution U(1,6)?

    <p>3.5</p> Signup and view all the answers

    In a uniform distribution U(1,6), what is the probability that the outcome will be 3?

    <p>$\frac{1}{6}$</p> Signup and view all the answers

    What does the moment generating function (MGF) of a random variable X provide?

    <p>Way to derive moments of the distribution</p> Signup and view all the answers

    In a uniform distribution, if all outcomes have equal probability, what type of distribution is it?

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

    "U(a,b)" notation is used for which type of distribution?

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

    What is the formula for calculating the variance of a uniform distribution?

    <p>$\frac{1}{12}(b - a + 1)^2$</p> Signup and view all the answers

    "E[X]" represents which statistical measure for a random variable X?

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

    What is the formula for the mean of a binomial distribution B(n,p)?

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

    What does the Poisson distribution describe?

    <p>Number of events in a fixed interval of time or space</p> Signup and view all the answers

    The moment generating function (MGF) for a binomial distribution can be derived using the formula:

    <p>$MX(t)=E[e^{tX}]$</p> Signup and view all the answers

    In the Poisson distribution, what is the variance formula?

    <p>$\lambda$</p> Signup and view all the answers

    For a continuous uniform distribution, what is the characteristic of every value between the range?

    <p>Equal likelihood of occurring</p> Signup and view all the answers

    What does the binomial distribution describe?

    <p>Number of successes in sequential Bernoulli trials</p> Signup and view all the answers

    What does the function $MX(t)=e^{2(et-1)}$ represent?

    <p>$MX(t)$ for Poisson distribution</p> Signup and view all the answers

    What is the Probability Mass Function (PMF) for the Poisson distribution?

    <p>!P(X=k)=k!e^-k/λ</p> Signup and view all the answers

    What is used to describe the number of events in a Poisson distribution?

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

    What does the moment generating function(MGF) describe?

    <p>Function to derive mean and variance.</p> Signup and view all the answers

    Study Notes

    Binomial Distribution

    • The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.
    • The probability mass function (PMF) is given by P(X=k) = (kn?)pk(1?p)n?k.
    • The mean (μ) is given by μ = np.
    • The variance (σ²) is given by σ² = np(1-p).
    • The moment generating function (MGF) is given by MX?(t) = ∑k=0n ekt(kn?)pk(1?p)n?k.

    Example: Binomial Distribution B(5,0.3)

    • Mean: μ = 5 × 0.3 = 1.5.
    • Variance: σ² = 5 × 0.3 × (1-0.3) = 1.05.
    • Moment Generating Function: MX?(t) = (0.7 + 0.21e^t + 0.0153e^2t).

    Poisson Distribution

    • The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space.
    • The probability mass function (PMF) is given by P(X=k) = k!e^(-λ?k!).
    • The mean (μ) is given by μ = λ.
    • The variance (σ²) is given by σ² = λ.
    • The moment generating function (MGF) is given by MX?(t) = e^λ(e^t - 1).

    Example: Poisson Distribution Poisson(2)

    • Mean: μ = 2.
    • Variance: σ² = 2.
    • Moment Generating Function: MX?(t) = e^2(e^t - 1).

    Continuous Uniform Distribution

    • A continuous uniform distribution is a probability distribution where every value between a certain range has an equal likelihood of occurring.

    Uniform Distribution (Discrete)

    • The uniform distribution is a discrete probability distribution where all outcomes have equal probability.
    • The probability mass function (PMF) is given by P(X=x) = 1/(b-a+1) for x=a,a+1,...,b.
    • The mean (μ) is given by μ = (a+b)/2.
    • The variance (σ²) is given by σ² = ((b-a+1)²-1)/12.
    • The moment generating function (MGF) is given by MX?(t) = ((b-a+1)(e^t - 1))/(e^(a+1)t - e^(b+1)t).

    Example: Uniform Distribution U(1,6)

    • Mean: μ = (1+6)/2 = 3.5.
    • Variance: σ² = ((6-1+1)²-1)/12 = 35/12.
    • Moment Generating Function: MX?(t) = (6(e^t - 1))/(e^(7t) - e^(2t)).

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    Description

    Learn about the properties of a uniform distribution where all outcomes have equal probability, denoted as U(a,b). Explore the probability mass function (PMF), mean, variance, and moment generating functions of a discrete uniform distribution.

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