The distribution of the difference between two independent normal random variables is also a normal distribution. The mean of this distribution is the difference of the means of the two original distributions, and the variance is the sum of the variances of the two original distributions.

Given that X1 and X2 are independent normal random variables with means 5 and 5.5 respectively, and both have a standard deviation of 1.5, the distribution of X1 - X2 will be a normal distribution with:

Mean = Mean of X1 - Mean of X2 = 5 - 5.5 = -0.5

Variance = Variance of X1 + Variance of X2 = (1.5)^2 + (1.5)^2 = 4.5

So, X1 - X2 ~ N(-0.5, 4.5)

To compute the probability that the time needed to construct the second device (X2) is less than the one needed to construct the first device (X1), we need to find the probability that X1 - X2 0.

We can standardize this to Z = (X1 - X2 - Mean) / sqrt(Variance) = (0 - (-0.5)) / sqrt(4.5) = 0.5 / sqrt(4.5) = 0.2357

Then, we look up this Z-score in the standard normal distribution table, or use a calculator or software to find the probability. The result is approximately 0.5934, or 59.34%.

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The distribution of the difference between two independent normal random variables is also a normal distribution. The mean of this distribution is the difference of the means of the two original distributions, and the variance is the sum of the variances of the two original distributions.

Given that X1 and X2 are independent normal random variables with means 5 and 5.5 respectively, and both have a standard deviation of 1.5, the distribution of X1 - X2 will be a normal distribution with:

Mean = Mean of X1 - Mean of X2 = 5 - 5.5 = -0.5

Variance = Variance of X1 + Variance of X2 = (1.5)^2 + (1.5)^2 = 4.5

So, X1 - X2 ~ N(-0.5, 4.5)

To compute the probability that the time needed to construct the second device (X2) is less than the one needed to construct the first device (X1), we need to find the probability that X1 - X2 > 0.

We can standardize this to Z = (X1 - X2 - Mean) / sqrt(Variance) = (0 - (-0.5)) / sqrt(4.5) = 0.5 / sqrt(4.5) = 0.2357

Then, we look up this Z-score in the standard normal distribution table, or use a calculator or software to find the probability. The result is approximately 0.5934, or 59.34%.

Knowee AI · Accepted Answer

The distribution of the difference between two independent normal random variables is also a normal distribution. The mean of this distribution is the difference of the means of the two original distributions, and the variance is the sum of the variances of the two original distributions.

Given that X1 and X2 are independent normal random variables with means 5 and 5.5 respectively, and both have a standard deviation of 1.5, the distribution of X1 - X2 will be a normal distribution with:

Mean = Mean of X1 - Mean of X2 = 5 - 5.5 = -0.5

Variance = Variance of X1 + Variance of X2 = (1.5)^2 + (1.5)^2 = 4.5

So, X1 - X2 ~ N(-0.5, 4.5)

To compute the probability that the time needed to construct the second device (X2) is less than the one needed to construct the first device (X1), we need to find the probability that X1 - X2 > 0.

We can standardize this to Z = (X1 - X2 - Mean) / sqrt(Variance) = (0 - (-0.5)) / sqrt(4.5) = 0.5 / sqrt(4.5) = 0.2357

Then, we look up this Z-score in the standard normal distribution table, or use a calculator or software to find the probability. The result is approximately 0.5934, or 59.34%.

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