To find the covariance between $X$ and $Y$, where $Y = e^X$ and $X \sim N(0,1)$, we can use the definition of covariance:

[ \text{Cov}(X,Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] ]

We are given that $X \sim N(0,1)$, meaning the mean of $X$ is 0 and the variance is 1. Thus, we need to calculate $\mathbb{E}[XY]$ and $\mathbb{E}[Y]$.

Step 1: Compute $\mathbb{E}[X]$

Since $X \sim N(0,1)$, we know: $\mathbb{E}[X] = 0$

Step 2: Compute $\mathbb{E}[Y]$

Since $Y = e^X$, we have: [ \mathbb{E}[Y] = \mathbb{E}[e^X] ] For a standard normal variable $X \sim N(0,1)$, we use the moment-generating function of the normal distribution, which for $X \sim N(0,1)$ is: $\mathbb{E}[e^X] = e^{\frac{1}{2}}$ Thus: $\mathbb{E}[Y] = e^{\frac{1}{2}}$

Step 3: Compute $\mathbb{E}[XY]$

We want to calculate $\mathbb{E}[XY]$, which is: [ \mathbb{E}[XY] = \mathbb{E}[X e^X] ] This requires integrating $x e^x$ over the probability density function (pdf) of $X$. The pdf of $X \sim N(0,1)$ is: $f_X(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ Thus: $\mathbb{E}[XY] = \int_{-\infty}^{\infty} x e^x \cdot \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} \, dx$ Simplifying the integrand: $\mathbb{E}[XY] = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} x e^{x - \frac{x^2}{2}} \, dx$ This is a standard result, and we know that: $\mathbb{E}[XY] = 0$ because $x e^{x - \frac{x^2}{2}}$ is an odd function and integrates to zero over the symmetric limits $-\infty$ to $\infty$.

Step 4: Calculate Covariance

Now we can use the covariance formula: [ \text{Cov}(X, Y) = \mathbb{E}[XY] - \mathbb{E}[X] \mathbb{E}[Y] ] Substitute the values we calculated: $\text{Cov}(X, Y) = 0 - (0 \cdot e^{\frac{1}{2}}) = 0$

Thus, the covariance between $X$ and $Y$ is:

$\boxed{0}$

Summary

The covariance between $X$ and $Y = e^X$, where $X \sim N(0,1)$, is $0$.

---

Would you like further clarification or any additional details on this problem? Here are a few related questions you may find interesting:

1. How do we calculate the covariance between a normal random variable and a nonlinear function of it?
2. What is the correlation coefficient between $X$ and $Y$?
3. How would the covariance change if $X \sim N(\mu, \sigma^2)$ instead of $N(0,1)$?
4. How can we compute $\mathbb{E}[Y^2]$ for $Y = e^X$?
5. What are the implications of zero covariance between two variables in a regression context?

Tip: Always remember that covariance measures the linear relationship between two variables, and if the covariance is zero, it suggests that there is no linear relationship, but it does not rule out non-linear relationships.