Let's solve Question 2 from the provided image step-by-step.

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Question 2(a):

The problem asks to find the distribution for which the characteristic function is:

$\phi(t) = e^{-\frac{1}{2} t^2 \sigma^2}.$

Solution:

The given characteristic function resembles the standard form of the Normal (Gaussian) distribution:

$\phi(t) = e^{i\mu t - \frac{1}{2} \sigma^2 t^2}.$

In the question, $\phi(t)$ does not have the $i\mu t$ term, implying the mean $\mu = 0$. Hence, the distribution is Normal with mean 0 and variance $\sigma^2$, i.e.,

$X \sim N(0, \sigma^2).$

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Question 2(b):

Find the mean, standard deviation, and harmonic mean of the Beta distribution of the first kind.

Beta Distribution (First Kind) Definition:

The probability density function (PDF) of the Beta distribution of the first kind, $B_1(\mu, \nu)$, is given by:

$f(x; \mu, \nu) = \frac{x^{\mu-1} (1-x)^{\nu-1}}{B(\mu, \nu)}, \quad 0 < x < 1,$

where $B(\mu, \nu)$ is the Beta function: $B(\mu, \nu) = \int_0^1 t^{\mu-1} (1-t)^{\nu-1} dt.$

1. Mean ($E[X]$): The mean of the Beta distribution is given by: $E[X] = \frac{\mu}{\mu + \nu}.$

2. Variance ($Var[X]$): The standard deviation is the square root of the variance. Variance for the Beta distribution is: $Var[X] = \frac{\mu \nu}{(\mu + \nu)^2 (\mu + \nu + 1)}.$ Thus, the standard deviation ($\sigma$) is: $\sigma = \sqrt{\frac{\mu \nu}{(\mu + \nu)^2 (\mu + \nu + 1)}}.$

3. Harmonic Mean: The harmonic mean of the Beta distribution is defined as: $HM[X] = \frac{\mu - 1}{\mu + \nu - 1}, \quad \text{for } \mu > 1.$

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Final Answers:

(a) The distribution is Normal distribution with mean $0$ and variance $\sigma^2$.

(b) For the Beta distribution:

• Mean: $\frac{\mu}{\mu + \nu}$,
• Standard Deviation: $\sqrt{\frac{\mu \nu}{(\mu + \nu)^2 (\mu + \nu + 1)}}$,
• Harmonic Mean: $\frac{\mu - 1}{\mu + \nu - 1}$ (valid for $\mu > 1$).

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Would you like a step-by-step derivation of any part of the solution?

Related Questions:

1. How do you derive the characteristic function of a normal distribution?
2. What is the difference between Beta distribution (first kind) and Beta prime (second kind)?
3. How is the Beta function related to the Gamma function?
4. Can you explain how to calculate moments of a Beta distribution?
5. What are the applications of Beta distributions in statistics?

Tip:

Remember that the Beta distribution is highly flexible and is widely used as a prior in Bayesian statistics due to its bounded support $[0,1]$.