The sum of two normally distributed random variables X and Y isa)Normally distributed, only if X and Y have the same standard deviationb)Always normally distributedc)Normally distributed, only if X and Y have the same meand)Normally distributed, only if X and Y are independentCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:

The sum of two normally distributed random variables X and Y is always normally distributed, regardless of whether X and Y have the same standard deviation, mean, or whether they are independent. This property is known as the additivity of normal distributions.

Proof:

Let X and Y be two normally distributed random variables. We can define X and Y in terms of their means (μ₁ and μ₂) and standard deviations (σ₁ and σ₂) as follows:

X ~ N(μ₁, σ₁)
Y ~ N(μ₂, σ₂)

Now, let's define a new random variable Z as the sum of X and Y:

Z = X + Y

To find the distribution of Z, we need to calculate its mean and standard deviation.

Mean:
The mean of Z can be calculated as the sum of the means of X and Y:

E(Z) = E(X + Y) = E(X) + E(Y) = μ₁ + μ₂

Standard Deviation:
The variance of Z can be calculated as the sum of the variances of X and Y:

Var(Z) = Var(X + Y) = Var(X) + Var(Y) = σ₁² + σ₂²

Therefore, the standard deviation of Z is the square root of the variance:

σ(Z) = sqrt(σ₁² + σ₂²)

Since Z has a mean of μ₁ + μ₂ and a standard deviation of sqrt(σ₁² + σ₂²), we can conclude that Z follows a normal distribution:

Z ~ N(μ₁ + μ₂, sqrt(σ₁² + σ₂²))

Conclusion:

The sum of two normally distributed random variables X and Y is always normally distributed, regardless of whether X and Y have the same standard deviation, mean, or whether they are independent. This property is a fundamental result of probability theory and is widely used in various fields such as statistics, finance, and engineering.

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