Question

An automobile dealership records the number of cars sold each day. The data are used in calculating the following probability distribution of daily sales:







a. Find the probability that the number of cars sold tomorrow will be between two and four (both inclusive).
b. Find the cumulative distribution function of the number of cars sold per day.
c. Show that P(x) is a probability distribution.

Answer 1

$$\begin{array}{c:c} x &Pr(X=x) \\ \hdashline 0 & 0.1 \\ \hdashline 1 & 0.1\\ \hdashline 2 & 0.2\\ \hdashline 3 & 0.3\\ \hdashline 4 & 0.2 \\ \hdashline 5 & 0.1 \end{array}$$

a)

$$Pr(2\leq X\leq4)=Pr(X=2)+Pr(X=3)+Pr(X=4)=$$

$$=0.2+0.3+0.2=0.7$$

b)

$$F(x)=Pr(X\leq x),$$

$Pr(X<0)=0,$

$Pr(X\leq0)=0.1,$

$Pr(X\leq1)=0.1+0.1=0.2,$

$Pr(X\leq2)= 0.2+0.2=0.4,$

$Pr(X\leq3)= 0.4+0.3=0.7,$

$Pr(X\leq4)= 0.7+0.2=0.9,$

$Pr(X\leq5)=0.9+0.1=1$

$$F(x) = \begin{cases} 0 & x<0 \\ 0.1 & 0\leq x<1 \\ 0.2 & 1\leq x<2\\ 0.4 & 2\leq x<3\\ 0.7 & 3\leq x<4\\ 0.9 & 4\leq x<5 \\ 1 & x\geq5 \end{cases}$$

c)

Each probability P(x) must be between 0 and 1

$$0\leq P(x)\leq 1$$

The sum of all the probabilities is 1

$$\sum P(x)=0.1+0.1+0.2+0.3+0.2+0.1=1$$

The P(x) represents the probability distribution of the random variable X.