Estimating lambda parameter

Question

I have random variable with exponential distribution. I am to find parameter Lambda. All i know is

$$\sum_{1}^{20}X_i^{2} = 30$$

I know i can solve this using moment method. Howeve , i fail to see the neccessarry steps in order to find

$$1/\lambda^{2} = 1/n-1 * \sum_{1}^{20} ( X_i - X_n ) ^{2}$$

I know i can expand right side to

$$\sum_{1}^{20} X_i^{2} - \sum_{1}^{2}2X_iX_n + \sum_{1}^{20}X_n^{2}$$

But thats where i am stuck. How to proceed with this calculation to find lambda?

All i see is that i can subtitute $\sum_{1}^{20} X_i^{2}$ with 30.

Thanks for help

Answer 1

Comment continued: Just to illustrate the suggestion. (Not a proof.) If this makes sense to you, then you can fill in the analytic details. Suppose $n=20,$ $\mu=\sqrt{3},$ $\mu^2 = 3.$ Then let's see what happens with a million random samples of size 20 from an exponential distribution with mean $\mu.$ Denote $Q = \frac 1n\sum_i X_i^2.$ (Computation using R statistical software.)

m = 10^6;  n = 20;  mu = sqrt(3);  lam=1/mu;  x = rexp(m*n, lam)
MAT=matrix(x, nrow=m)       # m x n matrix, each row a sample of 20
a = rowMeans(MAT);  mean(a)
## 1.731704                 # aprx E(X_i) = sqrt(3)
q = rowMeans(MAT^2);  mean(q)
## 5.996244                 # aprx E(Q) = 6