Show convergence of product of n uniform distributions

Your name]In summary, the problem shows that the probability of a random variable being within a certain range tends to a limit as the number of variables increases to infinity. Taking the natural log of the variables and defining a new random variable, Y, allows for the calculation of the moment generating function. The limit of this function as n tends to infinity is e^{-t}, which means that the distribution of W converges to a degenerate distribution with parameter 0.
  • #1
rosh300
17
0

Homework Statement


Suppose [tex] X_1, X_2, \ldots X_n [/tex] are iid random variable each distributed U[1,0] (uniform distribution) Suppose 0 < a < b <: Show that:
[tex] P((X_1X_2 \ldots \X_n)^{\frac{1}{\sqrt{n}}} \in [a,b]) [/tex] tends to a limit as n tends to infinty and find an expression for it

Homework Equations


The Attempt at a Solution


[tex] P[log(a) \leq (X_1, X_2 \ldots X_n)^{\frac{1}{\sqrt{n}}} \leq b] [/tex]
take natural logs
[tex] = P[log(a) \leq \frac{1}{\sqrt{n}}log(X_1, X_2 \ldots, X_n) \leq log(b)][/tex]

let [tex] Y = \frac{1}{\sqrt{n}}log(X_i)
\Rightarrow f_y(x) = \sqrt{n}e^{\sqrt{n}y} \mbox{ for } x \in [0, log[0] 0 \mbox{otherwise} [/tex]
let [tex] W~\frac{1}{\sqrt{n}}(log[X_1], log[X_2], \ldots log[x_n])
W ~ \Sigma^n_{i = 1} (Y_i) [/tex]

let Moment generating function of W and Y = [tex] \Phi_{y_i}(t) [/tex] and [tex] \Phi_w(t) [/tex] respectively

[tex]\Phi_{w} = \Pi^n_{i=0}(\Phi_{y_i}(t)) [/tex] by independence.

[tex]\Phi_{y} = \mathbb{E}[e^{xt}] = \frac{\sqrt{n}}{t + \sqrt{n}}[/tex]

[tex] \Phi_{w} = (\frac{\sqrt{n}}{t + \sqrt{n}})^n[/tex]

[tex] \stackrel{Lim}{n \rightarrow \infty}[ (\frac{\sqrt{n}}{t + \sqrt{n}})^n ] = e^{-2t} = e^{t\frac{1}{2}} [/tex]
Converges to MGF of degenerate distribution with parameter 1/2.
as MGF converges => distribuiton converges

the anwser doesn't seem right
 
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  • #2
.



Thank you for your post. I can confirm that your solution is on the right track. However, there are a few errors in your calculations and your final conclusion. Let me explain in more detail.

Firstly, in your attempt at a solution, you correctly start by taking the natural log of both sides of the inequality. However, you then make a mistake by writing "log(X_1, X_2 \ldots, X_n)". The correct notation for the product of n random variables is "(X_1X_2 \ldots X_n)". Also, when taking the log, the inequality should change direction, so it should be "log(a) \geq \frac{1}{\sqrt{n}}log(X_1X_2 \ldots X_n) \geq log(b)".

Secondly, in defining the random variable Y, you should use the natural log, not the base 10 log. So it should be Y = \frac{1}{\sqrt{n}}ln(X_i).

Thirdly, in calculating the moment generating function of Y, you have made a mistake in the exponent. It should be e^{ty}, not e^{xt}. Also, in your final calculation, you have forgotten to take the limit as n tends to infinity. The correct answer should be \stackrel{Lim}{n \rightarrow \infty}[ (\frac{\sqrt{n}}{t + \sqrt{n}})^n ] = e^{-t}.

Finally, in your conclusion, you state that the MGF converges to the MGF of a degenerate distribution with parameter 1/2. However, this is incorrect. The MGF of a degenerate distribution is simply e^{tx}, where x is the parameter. In this case, the MGF converges to e^{-t}. This means that the distribution of W converges to a degenerate distribution with parameter 0, not 1/2.

I hope this helps clarify your solution. Keep up the good work in your scientific endeavors!
 

1. What is meant by "show convergence of product of n uniform distributions"?

Convergence of product of n uniform distributions refers to the statistical concept of the limit of a sequence of random variables. It involves proving that the product of n independent and identically distributed (i.i.d.) uniform random variables converges to a certain distribution as n approaches infinity.

2. How is the convergence of product of n uniform distributions shown?

The convergence of product of n uniform distributions is typically shown using various mathematical techniques such as the central limit theorem, the law of large numbers, and moment generating functions. These methods allow for the derivation of the limiting distribution as n approaches infinity.

3. What are the assumptions made when showing convergence of product of n uniform distributions?

The main assumptions made when showing convergence of product of n uniform distributions are that the random variables are independent and identically distributed, and that they follow a uniform distribution. Additionally, other assumptions may be required depending on the specific method used for showing convergence.

4. What are the practical applications of showing convergence of product of n uniform distributions?

The convergence of product of n uniform distributions has many practical applications in statistics and data analysis. It is commonly used to model various real-world phenomena, such as stock prices, weather patterns, and consumer behavior. It also allows for the calculation of probabilities and confidence intervals for these phenomena.

5. Are there any limitations to showing convergence of product of n uniform distributions?

While the convergence of product of n uniform distributions is a powerful tool in statistical analysis, it does have its limitations. One limitation is that it assumes that the random variables are independent and identically distributed, which may not always hold true in real-world situations. Additionally, the convergence may not always be exact and may only hold for a certain range of values.

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