Statistics

We need to determine the distribution of the sum , where , , and represent the marks obtained by students in Physics, Maths, and Statistics, respectively. These are normally distributed with means 50, 52, and 48, and standard deviations 13, 12, and 16, respectively.

1. Properties of Normal Distributions:
If , , and are independent normal random variables, their sum is also normally distributed. Specifically:
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We assume independence since the problem does not indicate any correlation between the subjects.

2. Mean of the Sum:
For normal random variables, the mean of the sum is the sum of the means:

3. Variance of the Sum:
Since , , and are independent, the variance of the sum is the sum of the variances:


Calculate each variance:
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Sum the variances:


The standard deviation of the sum is:

4. Distribution of the Sum:
Since , , and are normally distributed and independent, their sum is also normally distributed:

Final Answer:
The distribution of is normal with mean 150 and variance 569, i.e., .