Discrete Uniform Distribution Variance: Quiz and Flashcards

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

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

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

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

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

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

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

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

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

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

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

What does the Poisson distribution describe?

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

In the Poisson distribution, what is the variance formula?

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

What does the binomial distribution describe?

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

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

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

What does the moment generating function(MGF) describe?

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)).

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.