## Questions and Answers

What is the formula for the mean of a binomial distribution with parameters n and p?

Which distribution describes the number of events occurring assuming they happen with a known constant rate and independently of time since the last event?

What is the variance formula of a Poisson distribution?

In a continuous uniform distribution, what is the likelihood of any value within the range occurring?

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What does the moment generating function (MGF) describe for a probability distribution?

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Which distribution involves the number of successes in fixed independent trials with the same probability of success?

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What is the Probability Mass Function (PMF) of a binomial distribution for k successes in n trials?

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Which parameter denotes the average rate of occurrence of events in a Poisson distribution?

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What does the Taylor series for ex simplify to when calculating the Moment Generating Function for the Poisson distribution?

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If X follows a Poisson distribution, what does X represent?

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What is the probability mass function (PMF) for the uniform distribution U(a,b)?

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What is the formula for the variance of a Uniform Distribution U(a,b)?

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What does the Moment Generating Function (MGF) of a random variable X help derive?

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For a discrete random variable X, what does MX(t) represent in the Moment Generating Function equation?

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If X follows a Uniform Distribution U(2,10), what is its mean?

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What is the probability mass function (PMF) value when using a value outside the range of a Uniform Distribution U(a,b)?

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In the MGF equation, what does $e^{tX}$ represent?

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When calculating the MGF for a Uniform Distribution U(3,8), what should be the limits of summation?

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For a Uniform Distribution U(4,12), what is the formula for its variance?

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What does the moment generating function MX(t) help to calculate for a random variable X?

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## Study Notes

### Binomial Distribution

- The binomial distribution describes the number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success.
- Notation: B(n, p) where n is the number of trials and p is the probability of success.
- Probability Mass Function (PMF): P(X=k) = (kn?)pk(1-p)n-k
- Mean (μ): μ = np
- Variance (σ²): σ² = np(1-p)
- Moment Generating Function (MGF): MX(t) = ∑k=0n (etk)(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 + 0.21et + 0.0153e2t)

### Poisson Distribution

- The Poisson distribution describes the number of events occurring in a fixed interval of time or space, assuming these events occur with a known constant rate and independently of the time since the last event.
- Notation: Poisson(λ)
- Probability Mass Function (PMF): P(X=k) = (k!e^(-λ))λ^k
- Mean (μ): μ = λ
- Variance (σ²): σ² = λ
- Moment Generating Function (MGF): MX(t) = e^(λ(et - 1))

### Example: Poisson Distribution (Poisson(2))

- Mean: μ = 2
- Variance: σ² = 2
- Moment Generating Function: MX(t) = e^(2(et - 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.
- Notation: U(a, b)
- Mean (μ): μ = (a + b) / 2
- Variance (σ²): σ² = (b - a)² / 12
- Moment Generating Function (MGF): MX(t) = (e^(bt) - e^(at)) / (b - a)

### Discrete Uniform Distribution

- The uniform distribution is a discrete probability distribution where all outcomes have equal probability.
- Notation: U(a, b)
- Probability Mass Function (PMF): P(X=x) = 1 / (b - a + 1) for x = a, a+1, ..., b
- Mean (μ): μ = (a + b) / 2
- Variance (σ²): σ² = (b - a + 1)² / 12
- Moment Generating Function (MGF): MX(t) = ((e^(bt) - e^(at)) / (b - a + 1))

### Example: Uniform Distribution (U(1, 6))

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

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## Description

Test your knowledge on the properties of the uniform distribution in discrete probability. Explore concepts like mean, variance, and moment generating functions for the uniform distribution with minimum and maximum values. Practice calculating probabilities using the Probability Mass Function (PMF) of the uniform distribution.