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Let X1 be a normal random variable with mean µ1 and variance σ 2 1 , and let X2 be a normal random variable with mean µ2 and variance σ 2 2 . Assuming that X1 and X2 are independent, what is the distribution of X1 + X2? g

Respuesta :

Answer:

[tex]X_1 + X_2 \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )[/tex]

Step-by-step explanation:

We are given the following in the question:

[tex]X_1[/tex] is a random normal variable with mean and variance

[tex]\mu_1\\\sigma_1^2[/tex]

[tex]X_1 \sim N(\mu_1,\sigma_1^2)[/tex]

[tex]X_2[/tex] is a random normal variable with mean and variance

[tex]\mu_2\\\sigma_2^2[/tex]

[tex]X_2 \sim N(\mu_2,\sigma_2^2)[/tex]

[tex]X_1, X_2[/tex] are independent events.

Let

[tex]Z =X_1 + X_2[/tex]

Then, Z will have a normal distribution with mean equal to the sum of the two means and its variance equal the sum of the two variances.

Thus, we can write:

[tex]\mu = \mu_1 + \mu_2\\\sigma^2 = \sigma_1^2 + \sigma_2^2\\Z \sim (\mu_1 + \mu_2, \sigma_1^2 + \sigma_2^2 )[/tex]

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