Random variable conv. in prob. to c. How to find c?

• SithsNGiggles
In summary, the distribution of the sum of independent standard normal random variables is a Chi Square distribution with n degrees of freedom. Converging in probability to a constant, Wn has a Chi Square distribution with n degrees of freedom.
SithsNGiggles

Homework Statement

Let ##Y_1,...Y_n## be independent standard normal random variables.

What is the distribution of ##\displaystyle\sum_{i=1}^n{Y_i}^2## ?

Let ##W_n=\displaystyle\frac{1}{n}\sum_{i=1}^n {Y_i}^2##. Does ##W_n\xrightarrow{p}c## for some constant ##c##? If so, what is the value of ##c##?

The Attempt at a Solution

In response to the first question, I say that ##W_n## has a Chi Square distribution with ##n## degrees of freedom.

For the second, I use a theorem stated in my book that says if ##\text{Var}(\hat{\theta}_n)\to 0##, then ##\hat{\theta}_n\xrightarrow{p}\theta##.

I have that ##\text{Var}(W_n)=\dfrac{1}{n^2}\text{Var}\left(\displaystyle\sum_{i=1}^n {Y_i}^2\right)=\dfrac{2n}{n^2}=\dfrac{2}{n}##, which approaches ##0## as ##n\to\infty##. Thus ##W_n## indeed converges in probability to ##c##.

The problem I'm having now is with finding the value of ##c##. There don't appear to be any examples in my text that describe the process to finding the limiting probability (if that's the right term). Any help is appreciated

SithsNGiggles said:

Homework Statement

Let ##Y_1,...Y_n## be independent standard normal random variables.

What is the distribution of ##\displaystyle\sum_{i=1}^n{Y_i}^2## ?

Let ##W_n=\displaystyle\frac{1}{n}\sum_{i=1}^n {Y_i}^2##. Does ##W_n\xrightarrow{p}c## for some constant ##c##? If so, what is the value of ##c##?

The Attempt at a Solution

In response to the first question, I say that ##W_n## has a Chi Square distribution with ##n## degrees of freedom.

For the second, I use a theorem stated in my book that says if ##\text{Var}(\hat{\theta}_n)\to 0##, then ##\hat{\theta}_n\xrightarrow{p}\theta##.

I have that ##\text{Var}(W_n)=\dfrac{1}{n^2}\text{Var}\left(\displaystyle\sum_{i=1}^n {Y_i}^2\right)=\dfrac{2n}{n^2}=\dfrac{2}{n}##, which approaches ##0## as ##n\to\infty##. Thus ##W_n## indeed converges in probability to ##c##.

The problem I'm having now is with finding the value of ##c##. There don't appear to be any examples in my text that describe the process to finding the limiting probability (if that's the right term). Any help is appreciated

What is ##E(W_n)##?

1 person
Judging by ##W_n##'s distribution, that would be ##n##. Does that mean ##c=n##, and if so, is that always the case (that the mean is the constant I'm supposed to find)?

SithsNGiggles said:
Judging by ##W_n##'s distribution, that would be ##n##. Does that mean ##c=n##, and if so, is that always the case (that the mean is the constant I'm supposed to find)?
No, E(Wn) is not n.

haruspex said:
No, E(Wn) is not n.

Oh right, I forgot that ##W_n=\frac{1}{n}\sum\cdots##. Thanks! I meant to say ##E(W_n)=1##.

SithsNGiggles said:
Oh right, I forgot that ##W_n=\frac{1}{n}\sum\cdots##. Thanks! I meant to say ##E(W_n)=1##.
Right - so all done here?

1. What is a random variable in probability?

A random variable in probability is a variable that can take on different values randomly according to a probability distribution. It is often denoted by the letter X and can be either discrete (taking on a finite or countably infinite number of values) or continuous (taking on values within a range).

2. What does it mean for a random variable to converge in probability?

Convergence in probability means that as the sample size increases, the values of a sequence of random variables get closer and closer to a single fixed value. In other words, the probability of the random variable taking on a particular value approaches a specific limit as the number of observations increases.

3. How is the convergence of a random variable to a constant c determined?

The convergence of a random variable to a constant c can be determined by using the definition of convergence in probability. This involves finding the limit of the probability of the random variable being within a certain range of the constant c as the sample size increases. If the limit is equal to 1, then the random variable converges to c in probability.

4. What is the role of the Central Limit Theorem in finding c for a random variable?

The Central Limit Theorem states that for a sufficiently large sample size, the distribution of the sample means of any random variable will approach a normal distribution. This theorem is often used to find the value of c for a random variable by calculating the mean and standard deviation of the sample and using them to approximate the normal distribution.

5. Can a random variable converge to more than one value c?

No, a random variable can only converge to a single value c in probability. This is because the definition of convergence in probability requires the probability of the random variable taking on values within a certain range of c to approach a specific limit as the sample size increases. If the probability approaches different limits for different values of c, then the convergence of the random variable is undefined.

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