consider two random variable x1, x2, which are uniformly distributed in the triangle. now suppose that y1 y2

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23-11-2022
Mathematics

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consider two random variable x1, x2, which are uniformly distributed in the triangle. now suppose that y1 y2

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contexto1025 contexto1025 · Answer 1 · 2022-11-29T10:31:58-05:00

The difference between 2X₁ and X₁+X₂ is N(0,6σ²)

According to the Central Limit Theorem, any large sum of the independent, identically distributed(iid) random variables is approximately Normal.

We know very well that the Normal distribution is defined by two parameters, the mean is μ , and the variance is σ² and written as X=N(μ,σ²).

Using the properties of normal random variables, we get

if X=N(μ₁,σ₁²)and Y=N(μ₁,σ₁²) are two independent identically distributed random variables then

the sum of normal random variables is represented by

=>X+Y=N(μ₁+u₂,σ₁²+σ₂²)

and the difference of normal random variables is represented by

=>X-Y =N( μ₁-u₂,σ₁²+σ₂²)

When Z=aX+bY, the linear combination of X and Y is given by

=>Z=N(aμ₁+bu₂,a²σ₁²+b²σ₂²)

Similarly, When Z=aX , the product of X is given by

=>Z= N(aμ₁,a²σ₁²)

We need to find 2X₁

Thus, following the property of multiplication, we get

=>2X₁=N(2μ,2²σ²)

=>2X₁=N(2μ,4σ²)

and following the property of addition,

X₁+X₂=N(μ+μ,σ²+σ²)

And the difference between the two is given by

2X₁-(X₁+X₂)=N(2μ-2μ,2σ₁²+4σ₂²)

=>2X₁-(X₁+X₂)=N(0,6σ²)

Hence, the required difference is N(0,6σ²).

To know more about Normal distribution, visit here:

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(Complete question) is:

consider two random variable X₁, X₂, which are uniformly distributed in the triangle. now suppose that Y₁ ,Y₂ are two random variables and If X₁ = N(μ, σ₂) and X₂ =N(μ, σ₂) are iid normal random variables, then what is the difference between 2 X₁ and X₁ + X₂?