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Answer: The weighted average of N separate measurements is a simple function of x_{1}, x_{2}, \ldots x_{N}. Therefore, the uncertainty in x wow can be found by error propagation.To prove: The uncertainty in x waw is clcimed.Now,consider the weighted average of N separate measurements x, x_{2}, \ldots x, x_{N}, with uncer tainties \sigma_{1}, \sigma_{2}, \cdots \sigma_{N}.\begin{array}{l}\operatorname{Prob} x\left(x_{A}\right) 2 \frac{1}{\sigma_{B}} e^{-\left(x_{A}-x\right)^{2} / 2 \sigma_{A}^{2}}(1) \\\operatorname{Prob} x\left(x_{B}\right) 2 \frac{1}{\sigma_{B}} e^{\frac{\left(x_{B}-x\right)^{2}}{2 \sigma_{B}}}-(2) \\\operatorname{Prob}\left(x_{A}, x_{B}\right)=\operatorname{Probx}\left(x_{A}\right) \cdot \operatorname{Prob}\left(x_{B}\right) \\2 \frac{1}{\sigma_{A} \sigma_{B}} e^{\frac{-x^{2}}{2}}\end{array}Here,I is the convenient shorthand.Nowx^{2} (chisquared) for the exponent.\begin{array}{l}x^{2}=\left(\frac{x_{A}-x}{\sigma_{A}}\right)^{2}+\left(\frac{x_{B}-x}{\sigma_{B}}\right)^{2} \cdot \frac{2 x_{A}-x}{\sigma_{A}^{2}}+\frac{2 x_{B}-x}{\sigma_{B}^{2}}=0 \\x=\frac{\left(\frac{x_{A}}{\sigma^{2} A}+\frac{x_{B}}{\sigma^{2} B}\right)}{\left(\frac{1}{\sigma^{2} A}+\frac{1}{\sigma_{B}}\right)}\end{array} \therefore W_{A}=\frac{1}{\sigma_{A}^{2}}andW_{B}=\frac{1}{\sigma^{2} B}\therefore x_{\text {wav }}=\frac{W_{A} X_{A}+W_{B} X_{B}}{W_{A}+W_{B}}Weighted awerage isx_{\text {aur }}=\sum_{i=1}^{N} \frac{w_{i} x_{i}}{\sum_{i=1}^{N} w_{i}}Now the weights arew i=\frac{1}{\sigma_{i}^{2}}The uncertainty on xavg\sigma_{\operatorname{ang}}=\frac{1}{\sqrt{\sum_{i=1}^{N} w_{i}}}If \sigma_{1}=\sigma_{2}=\sigma_{3}=\cdots \cdots \sigma_{N}=\sigma_{\text {, }}then the weighted werage is the simple average.The uncertainty is \frac{\sigma}{\sqrt{n}} and theweighted average have small uncentainty. which goes into average.Hence proved. NOTE::As per the rules i can answered ONLY ONE QUESTION. I HOPE YOUR HAPPY WITH MY ANSWER....***PLEASE SUPPORT ME WITH YOUR RATING... ***PLEASE GIVE ME "LIKE"...ITS VERY IMPORTANT FOR ME NOW....PLEASE SUPPORT ME ....THANK YOU ...