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If X and Y are 2 independent normal variables with mean as 10 and 12 and SD as 3 and 4, then (X+Y) is normally distributed with
- a)mean = 22 and SD = 7.
- b)mean = 22 and SD = 25.
- c)mean = 22 and SD = 5.
- d)mean = 22 and SD = 49.
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
If X and Y are 2 independent normal variables with mean as 10 and 12 a...
Explanation:
Step 1: Finding the mean of (X - Y)
As X and Y are independent normal variables, the mean of (X - Y) will be equal to the difference of their means i.e.,
Mean of (X - Y) = Mean of X - Mean of Y = 10 - 12 = -2
Step 2: Finding the variance of (X - Y)
As X and Y are independent normal variables, the variance of (X - Y) will be equal to the sum of their variances i.e.,
Variance of (X - Y) = Variance of X + Variance of Y = 3^2 + 4^2 = 9 + 16 = 25
Step 3: Finding the standard deviation of (X - Y)
The standard deviation of (X - Y) will be the square root of its variance i.e.,
Standard deviation of (X - Y) = sqrt(25) = 5
Step 4: Finding the mean of (X + Y)
As X and Y are independent normal variables, the mean of (X + Y) will be equal to the sum of their means i.e.,
Mean of (X + Y) = Mean of X + Mean of Y = 10 + 12 = 22
Step 5: Finding the variance of (X + Y)
As X and Y are independent normal variables, the variance of (X + Y) will be equal to the sum of their variances i.e.,
Variance of (X + Y) = Variance of X + Variance of Y = 3^2 + 4^2 = 9 + 16 = 25
Step 6: Finding the standard deviation of (X + Y)
The standard deviation of (X + Y) will be the square root of its variance i.e.,
Standard deviation of (X + Y) = sqrt(25) = 5
Conclusion:
Therefore, (X + Y) is normally distributed with mean = 22 and SD = 5. Hence, the correct option is (c).
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If X and Y are 2 independent normal variables with mean as 10 and 12 and SD as 3 and 4, then (X+Y) is normally distributed witha)mean = 22 and SD = 7.b)mean = 22 and SD = 25.c)mean = 22 and SD = 5.d)mean = 22 and SD = 49.Correct answer is option 'C'. Can you explain this answer?
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