Conditional expectation involving uniform distribution

Question

Let $Z_i$ be a random variable that takes value 1 when $U_i\le\frac 14$, and is zero otherwise, where $U_i$~Unif$(0,1)$. The goal is to find the expected value of $Z_i$. My working thus far is as follows:

So $$Z_i = \begin{cases} 1, & 0\le U_i\le \frac 14 \\ 0, & \text{otherwise} \end{cases}$$ I believe this makes $Z_i$ uniformly distributed, when $U_i$ is less than or equal to $\frac 14$. However, I don't exactly know what this means. If $U_i$ is also uniform on this interval, how does that factor into the proceedings? I am hopelessly lost. I feel like the law of iterated expectations must be involved here, but I don't know how to work to it.

EDIT: If $Z_i$ is Bernoulli distributed, not uniformly distributed, then is its PMF simply $$p_Z (z) = \begin{cases} \frac 14, & \text {if $k=1$}\\ \\\frac 34, & \text{otherwise} \end{cases}$$

Answer 1

$Z_i$ is basically an indicator variable. It will have a value of $1$ with probability $p={1\over4}$ and a value of $0$ with probability $1-p={3\over4}$. The expected value of an indicator variable is the probability that it is equal to $1$.

$$E(Z_i)=1\cdot P(Z_i=1) + 0\cdot P(Z_i=0)=P(Z_i=1)$$

Answer 2

$Z_i$ isn't uniformly distributed. It takes the value $1$ with the probability $1/4$ and the value $0$ else. So $Z_i$ is Bernoulli-distributed.