Question

If Y1, Y2 are identically independently distributed as normal distribution with mean u

Answer 1

Let $Y_1$ and $Y_2$ are independent and identically distributed normal random variable with mean $\mu$ and variance $\sigma^2.$

It is known that a linear combination of independent normal random variables is also normally distributed.

If $Y_1$ and $Y_2$ are independent and identically distributed normal random variable with mean $\mu$ and variance $\sigma^2,$ then $U=\dfrac{1}{2}(Y_1-3Y_2)$ is also normally distributed. So, the mean and variance of $U$are,

$$\mu_U=E[U]=E[\dfrac{1}{2}(Y_1-3Y_2)]=\dfrac{1}{2}E[Y_1-3Y_2]$$

$$=\dfrac{1}{2}(E[Y_1]-3E[Y_2])=\dfrac{1}{2}(\mu-3\mu)=-\mu$$

$$\sigma^2_U=V[U]=V[\dfrac{1}{2}(Y_1-3Y_2)]$$

$$=\dfrac{1}{4}(V[Y_1]+(-3)^2V[Y_2])$$

$$=\dfrac{1}{4}(\sigma^2+9\sigma^2)=\dfrac{5}{2}\sigma^2$$

Therefore, $U$ follows a normal distribution with mean $-\mu$ and variance $\dfrac{5}{2}\sigma^2$

$$U\sim N\big(-\mu, \dfrac{5}{2}\sigma^2\big)$$