Q.A coin is tossed thrice. Let X be the number of observed heads. Find the cumulative distribution function of X.

The cumulative distribution function $F(x)=P(X\leq x)$ gives the probability that a random variable takes a value up to and including $x$. For a discrete variable it is built by cumulatively adding the p.m.f., $F(x)=\sum_{x_i\leq x}p_i$, producing a step function; for a continuous variable it is $F(x)=\int_{-\infty}^{x}f(t)\,dt$, a smooth function whose derivative gives back …