Question

Complete the PDF and answer the questions.

a. Find the probability that$x=2.$

b. Find the expected value.

Short answer

(a) $P\left(x=2\right)=0.1$.

(b) The expected value is$1.6$.

Step-by-step solution

Given information (part a)

Explanation

$X$	$P\left(x\right)$	$X.P\left(x\right)$
$0$	$0.3$	$0$
$1$	$0.2$	$0.2$
$2$	$0.1$	$0.2$
$3$	$0.4$	$1.2$
Total	$1$	$1.6$

$$\begin{gathered} P\left(x=0\right)+P\left(x=1\right)+P\left(x=2\right)+P\left(x=3\right)=1 \\ 0.3+0.2+P\left(x=2\right)+0.4=1 \\ P\left(x=2\right)=1-0.9 \\ P\left(x=2\right)=0.1 \end{gathered}$$

The sum of all probabilities for a random variable is$1$. By this concept, we got the Probability of$x=2$.

Given information (part b)

Explanation (part b)

$X$	$P\left(X\right)$	$X.P\left(x\right)$
$0$	$0.3$	$0$
$1$	$0.2$	$0.2$
$2$	$0.1$	$0.2$
$3$	$0.4$	$1.2$
Total	$1$	$1.6$

$$\begin{gathered} E\left(x\right)=\underset{}{\sum xP\left(X=x\right)} \\ E\left(x\right)=0\times P\left(x=0\right)+1\times P\left(x=1\right)+2\times P\left(x=2\right)+3\times P\left(x=3\right) \\ E\left(x\right)=0\times 0.3+1\times 0.2+2\times 0.1+3\times 0.4 \\ E\left(x\right)=1.6 \end{gathered}$$

Thus, the expected value is$1.6$.