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INSTRUCTIONS: Read each problem carefully. Write solutions neatly. NO SOLUTION, NO POINTS. Express answers in lowest terms.

Find the expected value and the standard deviation of each random variable.
a,

x
10
20
30

P(x)
0.10
0.70
0.20

x^(**)P(x)

b.

x
2
4
6
8

P(x)
0.30
0.20
0.10
0.40

x^(**)P(x)

INSTRUCTIONS: Read each problem carefully. Write solutions neatly. NO SOLUTION, NO POINTS. Express answers in lowest terms. $\newline$ $1$ . Find the expected value and the standard deviation of each random variable. $\newline$ a, $\newline$ \begin{tabular}{|c|c|c|c|} $\newline$ \hline $x$ & $10$ & $20$ & $30$ \\ $\newline$ \hline $P(x)$ & $0$ . $10$ & $0$ . $70$ & $0$ . $20$ \\ $\newline$ \hline $x^{} P(x)$ & & & \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ b. $\newline$ \begin{tabular}{|c|c|c|c|c|} $\newline$ \hline $x$ & $2$ & $4$ & $6$ & $8$ \\ $\newline$ \hline $P(x)$ & $0$ . $30$ & $0$ . $20$ & $0$ . $10$ & $0$ . $40$ \\ $\newline$ \hline $x^{} P(x)$ & & & & \\ $\newline$ \hline $\newline$ \end{tabular}

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Find the magnitude of a vector

Full solution

Q. INSTRUCTIONS: Read each problem carefully. Write solutions neatly. NO SOLUTION, NO POINTS. Express answers in lowest terms.

\newline

1

. Find the expected value and the standard deviation of each random variable.

\newline

a,

\newline

\begin{tabular}{|c|c|c|c|}

\newline

\hline

x

&

10

&

20

&

30

\\

\newline

\hline

P(x)

&

0

.

10

&

0

.

70

&

0

.

20

\\

\newline

\hline

x^{*} P(x)

& & & \\

\newline

\hline

\newline

\end{tabular}

\newline

b.

\newline

\begin{tabular}{|c|c|c|c|c|}

\newline

\hline

x

&

2

&

4

&

6

&

8

\\

\newline

\hline

P(x)

&

0

.

30

&

0

.

20

&

0

.

10

&

0

.

40

\\

\newline

\hline

x^{*} P(x)

& & & & \\

\newline

\hline

\newline

\end{tabular}

Calculate Expected Value Set A: Calculate the expected value for set $a$ . Multiply each value of $x$ by its corresponding probability $P(x)$ and sum the results. $\newline$ Calculation: $(10 \times 0.10) + (20 \times 0.70) + (30 \times 0.20) = 1 + 14 + 6 = 21$ .
Calculate Variance Set A: Calculate the variance for set a. Subtract the expected value from each $x$ , square the result, multiply by the corresponding $P(x)$ , and sum all. $\newline$ Calculation: $((10-21)^2 \times 0.10) + ((20-21)^2 \times 0.70) + ((30-21)^2 \times 0.20) = 121 \times 0.10 + 1 \times 0.70 + 81 \times 0.20 = 12.1 + 0.7 + 16.2 = 29$ .
Calculate Standard Deviation Set A: Calculate the standard deviation for set $a$ by taking the square root of the variance. $\newline$ Calculation: $\sqrt{29} \approx 5.39$ .
Calculate Expected Value Set B: Calculate the expected value for set $b$ . Multiply each value of $x$ by its corresponding probability $P(x)$ and sum the results. $\newline$ Calculation: $(2 \times 0.30) + (4 \times 0.20) + (6 \times 0.10) + (8 \times 0.40) = 0.6 + 0.8 + 0.6 + 3.2 = 5.2$ .
Calculate Variance Set B: Calculate the variance for set $b$ . Subtract the expected value from each $x$ , square the result, multiply by the corresponding $P(x)$ , and sum all. $\newline$ Calculation: $((2-5.2)^2 \times 0.30) + ((4-5.2)^2 \times 0.20) + ((6-5.2)^2 \times 0.10) + ((8-5.2)^2 \times 0.40) = 10.24 \times 0.30 + 1.44 \times 0.20 + 0.64 \times 0.10 + 7.84 \times 0.40 = 3.072 + 0.288 + 0.064 + 3.136 = 6.56$ .
Calculate Standard Deviation Set B: Calculate the standard deviation for set b by taking the square root of the variance. $\newline$ Calculation: $\sqrt{6.56} \approx 2.56$ .

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