Intereting Posts

Generators of $\mathbb{Z}$ and $\mathbb{Z}$ when $\mathbb{Z}$, $\mathbb{Z}$ are f.g.
Algebraic Curves and Second Order Differential Equations
Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$
Does Kolmogorov 0-1 law apply to every translation invariant event?
Product of two Gaussian PDFs is a Gaussain PDF, but Produt of two Gaussan Variables is not Gaussian
$1 + 1 + 1 +\cdots = -\frac{1}{2}$
Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent
Do the real numbers and the complex numbers have the same cardinality?
Why does cross product give a vector which is perpendicular to a plane
Equivalence of induced representation
Inverse Laplace Transform of $\bar p_D = \frac{K_0(\sqrts r_D)}{sK_0(\sqrts)}$
Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$?
How do I show that $a^p + b^p > (a + b)^p$?
differentiate log Gamma function
Center of Heisenberg group- Dummit and Foote, pg 54, 2.2

Say you have $k$ i.i.d. normal random variables with some mean $\mu$ and variance $\sigma^2$ and you multiply them all together. What is the density function of the result?

- Maximum Likelihood Estimator of parameters of multinomial distribution
- Probability to choose specific item in a “weighted sampling without replacement” experiment
- 100 prisoners and a lightbulb
- Double pendulum probability distribution
- How to prove inverse direction for correlation coefficient?
- Calculating probabilities over longer period of time
- Poisson Distribution of sum of two random independent variables $X$, $Y$
- Probability of picking an odd number from the set of naturals?
- Solution to Locomotive Problem (Mosteller, Fifty Challenging Problems in Probability)
- Sum of Bernoulli random variables with different success probabilities

Let $X_k = Z_1 Z_2 \cdots Z_k$, where $Z_i$ are iid normal r.v. with mean $\mu$ and variance $\sigma^2$. The precise density would be hard to come by, but moments are easy to compute:

$$

\mathbb{E}(X_k^r) = \mathbb{E}(Z_1^r) \mathbb{E}(Z_2^r) \cdots \mathbb{E}(Z_k^r) = \left(m_r(Z)\right)^k

$$

Here is some simulation in *Mathematica*:

```
ProductNormalHistogram[k_Integer?Positive, \[Mu]_, \[Sigma]_] :=
Block[{x},
Histogram[
RandomVariate[
TransformedDistribution[Array[x, k, 1, Times],
Array[x, k] \[Distributed]
ProductDistribution[{NormalDistribution[\[Mu], \[Sigma]
], k}]], 10^5], Automatic, "PDF", ImageSize -> 250]]
```

For the case of $\mu=0$ and $\sigma = 1$, look-up product-normal distribution for some analytic results.

I think this paper would interest you:

The Distribution of Products of Beta, Gamma and Gaussian Random VariablesM. D. Springer and W. E. Thompson

SIAM Journal on Applied Mathematics , Vol. 18, No. 4 (Jun., 1970), pp. 721-737Published by: Society for Industrial and Applied Mathematics

Article Stable URL: http://www.jstor.org/stable/2099424

**Hint–Elementary(!)**

Prove that:

For continuous random variables $X$ and $Y$ with joint density $f$, the density of $Z=XY$ is given by $$f_Z(z)=\int_{-\infty}^\infty \frac 1 {|x|} f\left(x,\frac z x\right) \mathrm{d} x$$

Can you generalize this to $k$ of them by induction?

But, for me this looks like a far fetched idea in that, this is more elementary. (I have never had to bother about more than $2$ for computing by hand.)

- Curve enclosing the maximum area
- Is my proof of the uniqueness of $0$ non-circular?
- Is this convergent or divergent: $\sum _{n=1}^{\infty }\:\frac{2n^2}{5n^2+2n+1}$?
- Show that in a discrete metric space, every subset is both open and closed.
- How many tries to get at least k successes?
- Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum
- Proving that the number of vertices of odd degree in any graph G is even
- Rewriting the time-independent Schrödinger equation for a simple harmonic oscillating potential in terms of new variables
- Complex Numbers vs. Matrix
- Are $R=K/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K$ isomorphic?
- Calculation of $\lim_{x\rightarrow 0}\frac{\sin (\pi\cos^2 x)}{x^2}$
- Does absolute convergence of a sum imply uniform convergence?
- why binary is read right to left
- Consecutive coupon collection
- Schwarz Lemma – like exercise