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julsineya [31]
3 years ago
11

A certain brand of refrigerator has a useful life that is normally distributed with mean 10 years and standard deviation 3 years

. The useful lives of these refrigerators are independent. Calculate the probability that the total useful life of two randomly selected refrigerators will exceed 1.9 times the useful life of a third randomly selected refrigerator.
Mathematics
1 answer:
juin [17]3 years ago
3 0

Answer: 0.55567

Step-by-step explanation:

Given : A certain brand of refrigerator has a useful life that is normally distributed with mean 10 years and standard deviation 3 years. The useful lives of these refrigerators are independent.

i.e.

\mu=10\text{ years}\\\sigma=3\text{ years}

Let X_1 and X_2 are two randomly selected refrigerator's life whose sum will exceed third selected refrigerator X_3 .

So that, X =X_1+X_2-1.9\times X_3

Mean =E(X_1)+E(X_2)-E(X_3)=10+10-1.9\times10 =1

Standard deviation =\sqrt{(Var(X_1)+Var(X_1)+Var(X_1))}

=\sqrt{(3^2+3^2+(1.9\times3)^2)}

=7.1056315694

Z-score : z=\dfrac{X-E[x]}{\sqrt{Var[x]}}=\dfrac{0-1}{7.0156315694}\approx-0.14

Now , The  probability that the total useful life of two(i..e n=2) randomly selected refrigerators will exceed 1.9 times the useful life of a third randomly selected refrigerator would be :-

P(Z>-0.14)=P(Z

Hence, the required probability is 0.55567.

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