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# Discrete Uniform Distribution: Mean, Variance, Moment Generating Functions

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@SprightlyVision

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

• 1/5
• 1/7
• 1/8
• 1/6 (correct)
• ### What is the variance of a uniform distribution U(1,6)?

• 7/12 (correct)
• 4/3
• 5/3
• 3/7

• 0
• 2
• -1
• 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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