Let's compute the mean and variance for the given well-known distributions:

(i) Binomial Distribution (n, p):

• Mean: $\mu = n \cdot p$ where $n$ is the number of trials and $p$ is the probability of success in each trial.

• Variance: $\sigma^2 = n \cdot p \cdot (1 - p)$

(ii) Poisson Distribution (µ):

• Mean: $\mu = \mu$ where $\mu$ is the average rate of occurrence of an event.

• Variance: $\sigma^2 = \mu$

(iii) Normal Distribution (m, σ):

• Mean: $\mu = m$ where $m$ is the location parameter (mean) of the distribution.

• Variance: $\sigma^2 = \sigma^2$ where $\sigma$ is the standard deviation.

(iv) Gamma Distribution (l, k) with shape $k$ and rate $l$:

• Mean: $\mu = \frac{k}{\lambda}$ where $k$ is the shape parameter and $\lambda$ is the rate parameter.

• Variance: $\sigma^2 = \frac{k}{\lambda^2}$

(v) Cauchy Distribution (λ, µ):

• Mean: The mean for the Cauchy distribution is undefined. This is because the distribution does not have a well-defined expected value.

• Variance: The variance is also undefined, since the distribution's tails are heavy, making the variance infinite.

Summary:

• Binomial (n, p):

  • Mean: $n \cdot p$
  • Variance: $n \cdot p \cdot (1 - p)$

• Poisson (µ):

  • Mean: $\mu$
  • Variance: $\mu$

• Normal (m, σ):

  • Mean: $m$
  • Variance: $\sigma^2$

• Gamma (l, k):

  • Mean: $\frac{k}{\lambda}$
  • Variance: $\frac{k}{\lambda^2}$

• Cauchy (λ, µ):

  • Mean: Undefined
  • Variance: Undefined

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Would you like a deeper explanation on any of these? Let me know if you have any questions!

Here are 5 related questions:

1. How do you compute the mean and variance for a Geometric distribution?
2. What are the conditions for the Central Limit Theorem to hold?
3. How do you prove the mean of the Binomial distribution?
4. Can you explain the concept of moment-generating functions for distributions?
5. Why is the mean and variance undefined for the Cauchy distribution?

Tip: Variance gives you an idea of the spread of data in a distribution, and distributions with heavy tails, like the Cauchy distribution, often have undefined variance.