One weight, or six weights

Topic: Joint distributions · more questions on this topic
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The weights of people in a large population are modelled as normal with mean 75 kg and standard deviation 10 kg. Let X_1,\ldots,X_6 be the weights of six people drawn independently, and set Y_1=6X_1 and Y_2=X_1+\cdots+X_6.

What do Y_1 and Y_2 represent? Why do they have different distributions? Which has the smaller variance? State both distributions.

Worked answer (click to show)

Y_2 is the total weight of six people. Y_1 is six times the weight of one person. Both have mean 6\times 75=450. The variance of Y_1 is 36\times 100=3600. The variance of Y_2 is 6\times 100=600, six times smaller, because the six weights are independent. So Y_1\sim\operatorname{N}(450,3600) and Y_2\sim\operatorname{N}(450,600).