Chapter 11.3, Problem 10PPS

(a)

To determine

To find: The relative frequency table showing the theoretical probabilities.

Answer to Problem 10PPS

Therelative frequency table is shown in table 1

Explanation of Solution

Given:

The table is shown in Table 1

Table 1

Prize $x$	Winners
$100	1120
$250	800
$500	480
$1000	320
$2500	256
$5000	128
$7500	64
$10000	32

Calculation:

Consider the random variable X represents the amount prize by the players therefore the event space of X is,

  $X=\left\{100,250,100,2500,5000,7500,10000\right\}$

Consider the definition of the theatrical probability as,

  $P\left(X\right)=\frac{\text{total}\text{number}\text{of}\text{favourable}\text{event}}{\text{Total}\text{number}\text{of}\text{event}}$

Consider the frequency of $X=x_i$ is $f_i$ therefore the relative frequency or the theoretical probability is,

  $P\left(X=x_i\right)=\frac{f_i}{3200}$

Thus, the relative frequency table is shown in Table 1

Table 1

$X=x_i$	Number of winners	$P\left(X=x_i\right)$
$100	1120	0.35
$250	800	0.25
$500	480	0.15
$1000	320	0.10
$2500	256	0.08
$5000	128	0.04
$7500	64	0.02
$10000	32	0.01

(b)

To determine

To find: The graph for the theoretical probability distribution.

Answer to Problem 10PPS

Thegraph is shown in Figure 1

Explanation of Solution

From the theoretical data shown in Table 1 the graph for the data is shown in Figure 1

  

Figure 1

(c)

To determine

To find: The relative frequency table for 50 trials.

Answer to Problem 10PPS

The relative frequency table is shown in table 2

Explanation of Solution

Consider the probability that uses the same concept is mentioned as,

  $P\left(x_i\right)=\frac{f_i}{50}$

The required table is shown in Table 2

Table 2

$X=x_i$	Frequency	$P\left(X=x_i\right)$
$100	17	0.345
$250	12	0.247
$500	7	0.149
$1000	5	0.101
$2500	4	0.081
$5000	2	0.042
$7500	1	0.023
$10000	2	0.040

(d)

To determine

To find: The graph for the relative frequency table shown in Table 2

Answer to Problem 10PPS

Therequired graph is shown in Figure 2

Explanation of Solution

The graph for the table for 50 trials is shown in Figure 2

  

Figure 2

(e)

To determine

To find: The expected value of the random variable X .

Answer to Problem 10PPS

Thevalue of the random variable is $\$992.5$ .

Explanation of Solution

Consider the formula to determine the expected value of the random variable is,

  $E\left(X\right)={\displaystyle \sum _{i=1}^{n}P\left(X=x_i\right)}$

Then,

  $\begin{array}{c}E\left(X\right)=\left(\begin{array}{l}100\times 0.35+250\times 0.25+500\times 0.15+1000\times 0.1+2500\times 0.08\\ +5000\times 0.04+7500\times 0.02+10000\times 0.1\end{array}\right)\\ =\$992.5\end{array}$

(f)

To determine

To find: The standard deviation of the random variable X.

Answer to Problem 10PPS

The value of the standard deviation is $1713.34$ .

Explanation of Solution

Consider the formula to determine standard deviation of the random variable X is,

  $\sigma =\sqrt{{\displaystyle \sum _{i=1}^n\left[x_i-E\left(X\right)\right]P\left(x=x_i\right)}}$

Therefore,

  $\begin{array}{l}{\sigma }^2=\left({\left(100-922.5\right)}^2\times 0.35+{\left(250-992.5\right)}^2\times 0.25+\mathrm{....}{\left(10000-992.5\right)}^2\right)0.01\\ {\sigma }^2=2935518.75\\ \sigma =1713.34\end{array}$