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A certain job is completed in three steps in series. The means and standard deviations for the steps are (in minutes): Step 1 2 3 Mean 17 13 13 Standard Deviation 2 1 2

Assuming independent steps and normal distributions, compute the probability that the job will take less than 40 minutes to complete.

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Don't forget to accept answers to your previous questions if you find them fulfilling. –  Stefan Hansen Nov 13 '12 at 7:17
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1 Answer 1

If $X_1$, $X_2$, and $X_3$ are independent normals with means $\mu_1,\mu_2,\mu_3$ and variances $\sigma_1^2$, $\sigma_2^2,\sigma_3^2$, then $X_1+X_2+X_3$ is normally distributed, mean $\mu_1+\mu_2+\mu_3$ and variance $\sigma_1^2+ \sigma_2^2+\sigma_3^2$.

In our case, the sum has mean $43$ and variance $9$, so standard deviation $3$. Now all that remains is an ordinary normal distribution calculation. Note that $40$ is $1$ standard deviation below the mean.

Remark: A generalization can be useful. Under the same conditions of normality and independence, $\sum_{i=1}^n a_iX_i$ has normal distribution, mean $\sum_{i=1}^n a_i\mu_i$, and variance $\sum_{i=1}^n a_i^2\sigma_i^2$.

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does the a in the summation expression represent the numbers or? –  user48495 Nov 13 '12 at 6:18
    
@user48495: The stuff before the remark is the answer to your question. In the remark, I was addressing questions like $X_1,X_2,X_3,X_4$ are independent normals with known means and variances, and let $W$, for example, be $5X_1-17X_2+2.3X_3+\pi X_4$. what is the distribution of $W$? In your problem, all the $a_i$ are $1$, so I do not mention them. –  André Nicolas Nov 13 '12 at 6:23
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