## Questions and Answers

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)?

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

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

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

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

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

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

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

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What does the Poisson distribution describe?

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

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In the Poisson distribution, what is the variance formula?

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

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What does the binomial distribution describe?

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

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

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

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What does the moment generating function(MGF) describe?

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## 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.