Let $X_{1}$ and $X_{2}$ be two independently and identically distributed random variables with a common
$$f(x) = \frac{1}{\sqrt{2\pi}} \cdot \text{exp}(-x^{2})$$
for all $x \in (-\infty, \infty)$. Let $Y_{1} = aX_{1} + bX_{2}$ and $Y_{2} = cX_{1} + dX_{2}$ where parameters $a, b, c, d$ satisfy $a^2 + b^2 = 1$ and $c^2 + d^2 = 1$ and $ac + bd = 0$. Find the joint mgf of $(Y_{1}, Y_{2})$.
I need help solving this problem. So we want to find
$$\mathbb{E}[e^{sY_{1} + tY_{2}}] = \mathbb{E}[e^{s(aX_{1} + bX_{2})} \cdot e^{t(cX_{1} + dX_{2})}].$$
I don't think we can separate this into the product of two expectations because that would assume $Y_{1}$ and $Y_{2}$ are independent. Also, we can easily find the joint pdf of $X_{1}$ and $X_{2}$, which is just $f(x)^{2}$. I'm just not sure about how to do this problem, and I can't make use of the conditions.
EDIT: We can write
$$\mathbb{E}[e^{sY_{1} + tY_{2}}] = \mathbb{E}[e^{(sa + tc)X_{1}}] \cdot \mathbb{E}[e^{(sb + td)X_{2}}].$$
First we compute $\mathbb{E}[e^{(sa + tc)X_{1}}]$:
$$\mathbb{E}[e^{(sa + tc)X_{1}}] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{(sa + tc)x_{1} - x_{1}^{2}} \mathop{dx_{1}} $$
EDIT 2: Using an integration table, we get
$$\mathbb{E}[e^{(sa + tc)X_{1}}] = \frac{e^{(sa + tc)^{2}}}{\sqrt{2}}$$
Similarly,
$$\mathbb{E}[e^{(sb + td)X_{2}}] = \frac{e^{(sb + td)^{2}}}{\sqrt{2}}$$
So, the moment generating function is given by
$$\frac{e^{(sb + td)^{2}}}{\sqrt{2}} \frac{e^{(sa + tc)^{2}}}{\sqrt{2}} = \frac{e^{s^2b^2 + 2sbtd + t^2d^2 + s^2a^2 + 2satc + t^2c^2}}{2}$$
But, the exponent of $e$ can be rewritten as
$$s^2b^2 + 2sbtd + t^2d^2 + s^2a^2 + 2satc + t^2c^2 = s^{2}(a^{2} + b^{2}) + t^2(c^2 + d^2) + 2st(ac + bd) = s^2 + t^2. $$
So, our moment generating function is given by
$$e^{s^2 + t^2}/2 $$
We can directly use the indepencence of $X_1$ and $X_2$:
$$ E\left[e^{sY_1+tY_2}\right] = E\left[e^{(sa+tc)X_1}e^{(sb+td)X_2}\right] = E\left[e^{(sa+tc)X_1}\right]E\left[e^{(sb+td)X_2}\right]. $$
Now for example $(sa+tc)X_1 \sim N(0,(sa+tc)^2)$, so the first factor is the mean of a log-normal distribution with parameters $\mu = 0$ and $\sigma^2 = (sa+tc)^2$, which is $$ E\left[e^{(sa+tc)X_1}\right] = e^{\frac{(sa+tc)^2}{2}}. $$