Question

Let $p_X(x)$ be the pmf of a random variable $X.$ Find the cdf $F(x)$ of $X$ and sketch its graph along with that of $p_X(x)$ if: (a) $p_X(x)=1,x=0$, zero elsewhere. (b) $p_X(x)=\frac{1}{3},x=-1,0,1$, zero elsewhere. (c) $p_X(x)=x/15,x=1,2,3,4,5$, zero elsewhere.

Short answer

For scenario (a), $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$. For scenario (b), $F(x)=0$ for $x<-1$, $F(x)=\frac{1}{3}$ for $-1\le x<0$, $F(x)=\frac{2}{3}$ for $0\le x<1$, $F(x)=1$ for $x\ge 1$. For scenario (c), $F(x)=0$ for $x<1$, $F(x)=\frac{1}{15}$ for $1\le x<2$, $F(x)=\frac{3}{15}$ for $2\le x<3$, $F(x)=\frac{6}{15}$ for $3\le x<4$, $F(x)=\frac{10}{15}$ for $4\le x<5$, $F(x)=1$ for $x\ge 5$. The graph of the CDF in each case is a step function that jumps at the values of $x$ defined by the PMF, and the PMF graphs are vertical lines at these $x$ values.

Step-by-step solution

Scenario (a)

Here, $p_X(x)=1$ for $x=0$ and zero elsewhere. The CDF $F(x)$ is defined as probability that the variable takes a value less than or equal to x. So, $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ since $p_X(0)=1$. If you sketch the graph for $F(x)$ it would be a step function that jumps from 0 to 1 at $x=0$. The graph of $p_X(x)$ would be a vertical line at $x=0$.

Scenario (b)

For this case, $p_X(x)=\frac{1}{3}$ for $x=-1,0,1$ and zero elsewhere. The CDF is defined as: $F(x)=0$ for $x<-1$, $F(x)=\frac{1}{3}$ for $-1\le x<0$, $F(x)=\frac{2}{3}$ for $0\le x<1$, $F(x)=1$ for $x\ge 1$.This is because for each $x$, we are adding up all the probabilities less than or equal to that $x$. The graph of $F(x)$ would be a step function that increases by $\frac{1}{3}$ at $x=-1,0,1$. The graph of $p_X(x)$ would be vertical lines at $x=-1,0,1$.

Scenario (c)

In this case, $p_X(x)=\frac{x}{15}$ for $x=1,2,3,4,5$ and zero elsewhere. The CDF is then calculated by taking cumulative sum of the probabilities: $F(x)=0$ for $x<1$, $F(x)=\frac{1}{15}$ for $1\le x<2$, $F(x)=\frac{3}{15}$ for $2\le x<3$, $F(x)=\frac{6}{15}$ for $3\le x<4$, $F(x)=\frac{10}{15}$ for $4\le x<5$, $F(x)=1$ for $x\ge 5$.The graph of $F(x)$ would be a step function that increases by $\frac{x}{15}$ sequentially at each integer $x$ from 1 to 5. The graph of $p_X(x)$ would be vertical lines at $x=1,2,3,4,5$.