I have done the following question:

In a hospital, 1/3 of the people who use a particular elevator are adults and 2/3 are children. The weight of the adults follows a normal distribution with mean 70 kg and sd 18 kg. The weight of the children follows a normal distribution with mean 18 kg and sd 8kg. Calculate the probability that a person entering the elevator weighs more than 45 kg.

I did the following: 1/3 Pr(X > 45) + 2/3 Pr(Y > 45) = 0.311705

and got the right answer.

However, I also tried doing it by defining a new random variable W = X/3 + 2Y/3:

I found that W is a normal random variable with mean 40.66667 and sd 8.02773. Then I calculated Pr(W > 45) and got 0.294669 which is wrong. I don't know why this method is wrong. If someone could explain the error I'd be grateful. Is there some different way that the normal random variables get combined?

Thanks.

The reason I tried this second method is that another part of the question asks to calculate the probability that the weight of three people entering the lift will exceed 110 kg. I figured that I could just use the normal distribution corresponding to V = W + W + W and calculate Pr(V > 110). But this will be wrong if my definition of W is wrong ....

Any help will be appreciated.

In a hospital, 1/3 of the people who use a particular elevator are adults and 2/3 are children. The weight of the adults follows a normal distribution with mean 70 kg and sd 18 kg. The weight of the children follows a normal distribution with mean 18 kg and sd 8kg. Calculate the probability that a person entering the elevator weighs more than 45 kg.

I did the following: 1/3 Pr(X > 45) + 2/3 Pr(Y > 45) = 0.311705

and got the right answer.

However, I also tried doing it by defining a new random variable W = X/3 + 2Y/3:

I found that W is a normal random variable with mean 40.66667 and sd 8.02773. Then I calculated Pr(W > 45) and got 0.294669 which is wrong. I don't know why this method is wrong. If someone could explain the error I'd be grateful. Is there some different way that the normal random variables get combined?

Thanks.

The reason I tried this second method is that another part of the question asks to calculate the probability that the weight of three people entering the lift will exceed 110 kg. I figured that I could just use the normal distribution corresponding to V = W + W + W and calculate Pr(V > 110). But this will be wrong if my definition of W is wrong ....

Any help will be appreciated.

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