Consider a fair coin. Let $X:\{H,T\}\to \Bbb R$ be a random variable with $X(H)=-1$ and $X(T)=1$.
- Find the PDF of X $$P\{X\le t\} $$
- Flipping the coin 4 times and associate with the sequence of iid {Xn} where Xn are defined as above. Compute $$P\{\frac{X_1+...+X_4}{2}\le t\} $$
I am able to find $E(X)=0$ and $var(X)=1$. But I am not sure how to find the probability density function $f(x)$ for $P\{X\le t\}$, where $P\{X\le t\}=\int_\infty^tf(x)dx$. Since X can only have two values, do I need to discuss different values for t?
Any help would be really appreciated.
Since $X$ is a discrete random variable, with support of $\{-1,1\}$ and a probability mass function (pmf) of $\mathsf p_X(x)~=~\tfrac 1 2~\mathbf 1_{x\in\{-1;1\}}$ , then the Cumulative Distribution Function (CDF) will be a piecewise function. Also since this is a discrete random variable, we sum rather than integrate as we would for a continuous random variable.
$$F_X(t) ~=~ \sum_{\substack{x\in \{-1,1\}\\x\leq t}} \mathsf p_X(x)$$
The value will be zero for $t$ less than $-1$, and one for $t$ of at least $1$ , but what will it be for $t$ in the intervening interval?
$$F_X(t)~=~\begin{cases}0 & : t<-1\\ ? & : -1\leq t< 1\\1 & : 1\leq t\end{cases}$$
$\Box$