The random variable $x$ represents the number of tests that a patient entering a clinic will have along with the corresponding probabilities. Find the mean and standard deviation for the random variable $x$ . Compute and interpret the mean and standard deviation of a discrete random variable $x$ $\quad$ $P(x)$ $\quad$ $0$ $\quad$ $\frac{3}{17}$ $\quad$ $x$ $0$ $\quad$ $x$ $2$ $\quad$ $x$ $4$ $\quad$ $x$ $6$ $\quad$ $x$ $8$ $\quad$ $x$ $0$ $\quad$ $x$ $2$ $\quad$ $x$ $4$

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Grade 7

Calculate quartiles and interquartile range

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Q. The random variable

x

represents the number of tests that a patient entering a clinic will have along with the corresponding probabilities. Find the mean and standard deviation for the random variable

x

. Compute and interpret the mean and standard deviation of a discrete random variable

x

\quad

P(x)

\quad

0

\quad

\frac{3}{17}

\quad

x

0

\quad

x

2

\quad

x

4

\quad

x

6

\quad

x

8

\quad

x

0

\quad

x

2

\quad

x

4

Calculate Mean: Calculate the mean (expected value) of the random variable $x$ . To find the mean, we multiply each value of $x$ by its corresponding probability and sum the results. Mean ( $\mu$ ) = $\Sigma [x \cdot P(x)]$ = $(0 \cdot \frac{3}{17}) + (1 \cdot \frac{5}{17}) + (2 \cdot \frac{6}{17}) + (3 \cdot \frac{2}{17}) + (4 \cdot \frac{1}{17})$ = $(0) + (\frac{5}{17}) + (\frac{12}{17}) + (\frac{6}{17}) + (\frac{4}{17})$ = $(5 + 12 + 6 + 4) / 17$ = $\frac{27}{17}$
Calculate Variance: Calculate the variance of the random variable $x$ . To find the variance, we use the formula $\text{Var}(x) = \Sigma [(x - \mu)^2 \cdot P(x)]$ . Variance (\sigma^\(2) = ( $0$ - \frac{ $27$ }{ $17$ })^ $2$ \cdot (\frac{ $3$ }{ $17$ }) + ( $1$ - \frac{ $27$ }{ $17$ })^ $2$ \cdot (\frac{ $5$ }{ $17$ }) + ( $2$ - \frac{ $27$ }{ $17$ })^ $2$ \cdot (\frac{ $6$ }{ $17$ }) + ( $3$ - \frac{ $27$ }{ $17$ })^ $2$ \cdot (\frac{ $2$ }{ $17$ }) + ( $4$ - \frac{ $27$ }{ $17$ })^ $2$ \cdot (\frac{ $1$ }{ $17$ }) = (\frac{ $729$ }{ $289$ }) \cdot (\frac{ $3$ }{ $17$ }) + (\frac{ $64$ }{ $289$ }) \cdot (\frac{ $5$ }{ $17$ }) + (\frac{ $1$ }{ $289$ }) \cdot (\frac{ $6$ }{ $17$ }) + (\frac{ $256$ }{ $289$ }) \cdot (\frac{ $2$ }{ $17$ }) + (\frac{ $961$ }{ $289$ }) \cdot (\frac{ $1$ }{ $17$ }) = (\frac{ $729$ }{ $289$ }) \cdot (\frac{ $3$ }{ $17$ }) + (\frac{ $320$ }{ $289$ }) \cdot (\frac{ $5$ }{ $17$ }) + (\frac{ $6$ }{ $289$ }) \cdot (\frac{ $6$ }{ $17$ }) + (\frac{ $512$ }{ $289$ }) \cdot (\frac{ $2$ }{ $17$ }) + (\frac{ $961$ }{ $289$ }) \cdot (\frac{ $1$ }{ $17$ }) = (\frac{ $2187$ }{ $4913$ }) + (\frac{ $1600$ }{ $4913$ }) + (\frac{ $36$ }{ $4913$ }) + (\frac{ $1024$ }{ $4913$ }) + (\frac{ $961$ }{ $4913$ }) = (\frac{ $2187$ + $1600$ + $36$ + $1024$ + $961$ }{ $4913$ }) = \frac{ $5808$ }{ $4913$ }
Calculate Standard Deviation: Calculate the standard deviation of the random variable $x$ . The standard deviation is the square root of the variance. Standard deviation ( $\sigma$ ) = $\sqrt{\sigma^2}$ = $\sqrt{\frac{5808}{4913}}$ = $\sqrt{1.182}$ = $1.087$ (approximately)