# Related to Gamma distribution and chi square distribution

I have $2N$ independent normal random variables $X_1,X_2\cdots X_{2N}$ where every variable has mean zero and variance $\frac{1}{2}$. I want to find the cumulative distribution function of $T=X_1^2+X_2^2+\cdots X_{2N}^2$. I know that I can make $N$ exponential random variables (where every exponential random variable will have mean $1$) and the sum of these $N$ exponential random variables will follow a gamma distribution. Therefore the CDF will be $$\text{cdf of T}=\frac{\gamma(N,x)}{\Gamma(N)}$$. However, if I try to find it through chi square distribution then my answer does not matches with the above one. Here is my attempt with chi square distribution

My attempt with chi square distribution:

We know that $T=\left(\frac{1}{2}\right)\left[2\times X_1^2+2\times X_2^2+\cdots 2\times X_{2N}^2\right]$. Now every $2\times X_i^2$ is a standard normal random variable. Therefore the probability density function of $T$ will be (according to https://stats.stackexchange.com/questions/118676/relationship-between-gamma-and-chi-squared-distribution) $$\text{pdf of T}=\frac{1}{\sigma^2}\times f_{\chi^2}(\frac{x}{\sigma^2},2N)=2\times \frac{1}{2^N\Gamma(N)} x^{n-1}e^{-x}$$ and corresponding to this the CDF of $T$ is $$F_T(c)=\int_0^{\infty}\text{pdf of T }dx=\frac{1}{2^{N-1}\Gamma(N)}\gamma(N,c)$$ which is not exactly same as my very first equation above. Please guide where I am doing it wrong.

The $\chi^2$ distribution with $2N$ degrees of freedom is the distribution of the sum of squares of $2N$ standard normal random variables. Let $F_{2N}(x)$ be the cdf of that. In your case, the normal random variables $X_j$ have variance $1/2$ rather than $1$, so $X_j = Y_j/\sqrt{2}$ where $Y_j$ is a standard normal random variable, and $T = \sum_{j} X_j^2 = (1/2) \sum_j Y_j^2$. Thus the CDF for $T$ is

$$F_T(x) = P(T \le x) = P(2T \le 2x) = F_{2N}(2x)$$

• Thanks alot for your answer. Can you point the error in my attempt. I will be very thankful to you. Commented Aug 30, 2016 at 2:26
The text passage discusses the cumulative distribution function (CDF) of a sum of squared normal random variables and compares two methods of deriving the CDF: using the gamma distribution and using the chi-square distribution. The author presents a discrepancy between the two methods and asks for guidance on where the error might be.
single_question_with_answer

the passage contains a single mathematical question regarding the discrepancy between the cdf derived using the gamma distribution and the chi-square distribution. an answer is provided, explaining that the normal random variables have a variance of 1/2, and thus the correct transformation to the chi-square distribution involves scaling by 2.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Related to Gamma distribution and chi square distribution

I have $2N$ independent normal random variables $X_1,X_2\cdots X_{2N}$ where every variable has mean zero and variance $\frac{1}{2}$. I want to find the cumulative distribution function of $T=X_1^2+X_2^2+\cdots X_{2N}^2$. I know that I can make $N$ exponential random variables (where every exponential random variable will have mean $1$) and the sum of these $N$ exponential random variables will follow a gamma distribution. Therefore the CDF will be $$\text{cdf of T}=\frac{\gamma(N,x)}{\Gamma(N)}$$. However, if I try to find it through chi square distribution then my answer does not matches with the above one. Here is my attempt with chi square distribution

My attempt with chi square distribution:

We know that $T=\left(\frac{1}{2}\right)\left[2\times X_1^2+2\times X_2^2+\cdots 2\times X_{2N}^2\right]$. Now every $2\times X_i^2$ is a standard normal random variable. Therefore the probability density function of $T$ will be (according to https://stats.stackexchange.com/questions/118676/relationship-between-gamma-and-chi-squared-distribution) $$\text{pdf of T}=\frac{1}{\sigma^2}\times f_{\chi^2}(\frac{x}{\sigma^2},2N)=2\times \frac{1}{2^N\Gamma(N)} x^{n-1}e^{-x}$$ and corresponding to this the CDF of $T$ is $$F_T(c)=\int_0^{\infty}\text{pdf of T }dx=\frac{1}{2^{N-1}\Gamma(N)}\gamma(N,c)$$ which is not exactly same as my very first equation above. Please guide where I am doing it wrong.

The $\chi^2$ distribution with $2N$ degrees of freedom is the distribution of the sum of squares of $2N$ standard normal random variables. Let $F_{2N}(x)$ be the cdf of that. In your case, the normal random variables $X_j$ have variance $1/2$ rather than $1$, so $X_j = Y_j/\sqrt{2}$ where $Y_j$ is a standard normal random variable, and $T = \sum_{j} X_j^2 = (1/2) \sum_j Y_j^2$. Thus the CDF for $T$ is

$$F_T(x) = P(T \le x) = P(2T \le 2x) = F_{2N}(2x)$$

• Thanks alot for your answer. Can you point the error in my attempt. I will be very thankful to you. Commented Aug 30, 2016 at 2:26
user
The text passage discusses the cumulative distribution function (CDF) of a sum of squared normal random variables and compares two methods of deriving the CDF: using the gamma distribution and using the chi-square distribution. The author presents a discrepancy between the two methods and asks for guidance on where the error might be.

TYPE: SINGLE_QUESTION_WITH_ANSWER

The passage contains a single mathematical question regarding the discrepancy between the CDF derived using the gamma distribution and the chi-square distribution. An answer is provided, explaining that the normal random variables have a variance of 1/2, and thus the correct transformation to the chi-square distribution involves scaling by 2.