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Compute the mean and standard deviation of the random variable with the given discrete probability distribution.

x= -9, p(x) = 0.11
x= -7, p(x) =0.24
x= 0, p(x) = 0.26
x= 3, p(x) = 0.25
x=5, p(x) = 0.14

(a) Find the mean. Round the answer to three decimal places, if necessary.
(b) Find the standard deviation. Round the answer to three decimal places, if necessary.

5 answers

To compute the mean and standard deviation of the given discrete probability distribution, we will use the formulas for the mean (expected value) and standard deviation.

(a) Mean Calculation

The mean \( \mu \) of a discrete random variable is calculated using the formula:

\[ \mu = \sum (x \cdot p(x)) \]

Where \( x \) is the value of the random variable and \( p(x) \) is the probability of \( x \).

Given the data:

  • \( x = -9, p(x) = 0.11 \)
  • \( x = -7, p(x) = 0.24 \)
  • \( x = 0, p(x) = 0.26 \)
  • \( x = 3, p(x) = 0.25 \)
  • \( x = 5, p(x) = 0.14 \)

Calculating the mean:

\[ \mu = (-9 \cdot 0.11) + (-7 \cdot 0.24) + (0 \cdot 0.26) + (3 \cdot 0.25) + (5 \cdot 0.14) \]

Calculating each term:

  1. \( -9 \cdot 0.11 = -0.99 \)
  2. \( -7 \cdot 0.24 = -1.68 \)
  3. \( 0 \cdot 0.26 = 0 \)
  4. \( 3 \cdot 0.25 = 0.75 \)
  5. \( 5 \cdot 0.14 = 0.70 \)

Now summing them up:

\[ \mu = -0.99 - 1.68 + 0 + 0.75 + 0.70 = -1.22 \]

So, the mean is:

\[ \mu = -1.220 \]

(b) Standard Deviation Calculation

The standard deviation \( \sigma \) is calculated using the formula:

\[ \sigma = \sqrt{\sum (x - \mu)^2 \cdot p(x)} \]

First, we need the variance, which is:

\[ \sigma^2 = \sum (x - \mu)^2 \cdot p(x) \]

Calculating each term \( (x - \mu)^2 \):

  1. For \( x = -9 \): \[ (-9 - (-1.22))^2 = (-7.78)^2 = 60.7284 \] Contribution to variance: \( 60.7284 \cdot 0.11 = 6.679124 \)

  2. For \( x = -7 \): \[ (-7 - (-1.22))^2 = (-5.78)^2 = 33.4084 \] Contribution to variance: \( 33.4084 \cdot 0.24 = 8.049816 \)

  3. For \( x = 0 \): \[ (0 - (-1.22))^2 = (1.22)^2 = 1.4884 \] Contribution to variance: \( 1.4884 \cdot 0.26 = 0.387004 \)

  4. For \( x = 3 \): \[ (3 - (-1.22))^2 = (4.22)^2 = 17.8084 \] Contribution to variance: \( 17.8084 \cdot 0.25 = 4.4521 \)

  5. For \( x = 5 \): \[ (5 - (-1.22))^2 = (6.22)^2 = 38.6884 \] Contribution to variance: \( 38.6884 \cdot 0.14 = 5.414416 \)

Summing these contributions to variance:

\[ \sigma^2 = 6.679124 + 8.049816 + 0.387004 + 4.4521 + 5.414416 \approx 24.98246 \]

Now, take the square root to find the standard deviation:

\[ \sigma = \sqrt{24.98246} \approx 4.998 \]

Final Answers

(a) Mean \( \mu \) = -1.220

(b) Standard deviation \( \sigma \) = 4.998

A is correct but one of your digits for B is incorrect

Thank you for pointing that out. Let's re-calculate the standard deviation carefully step by step.

(b) Recalculation of Standard Deviation

The standard deviation \( \sigma \) is calculated using the formula:

\[ \sigma = \sqrt{\sum (x - \mu)^2 \cdot p(x)} \]

Where \( \mu = -1.22 \) is the mean we calculated earlier.

We need to compute each term \( (x - \mu)^2 \cdot p(x) \):

  1. For \( x = -9 \): \[ (-9 - (-1.22))^2 = (-7.78)^2 = 60.7284 \] Contribution to variance: \[ 60.7284 \cdot 0.11 = 6.679124 \]

  2. For \( x = -7 \): \[ (-7 - (-1.22))^2 = (-5.78)^2 = 33.4084 \] Contribution to variance: \[ 33.4084 \cdot 0.24 = 8.049816 \]

  3. For \( x = 0 \): \[ (0 - (-1.22))^2 = (1.22)^2 = 1.4884 \] Contribution to variance: \[ 1.4884 \cdot 0.26 = 0.387004 \]

  4. For \( x = 3 \): \[ (3 - (-1.22))^2 = (4.22)^2 = 17.8084 \] Contribution to variance: \[ 17.8084 \cdot 0.25 = 4.4521 \]

  5. For \( x = 5 \): \[ (5 - (-1.22))^2 = (6.22)^2 = 38.6884 \] Contribution to variance: \[ 38.6884 \cdot 0.14 = 5.414416 \]

Now we sum these contributions to calculate the variance:

\[ \sigma^2 = 6.679124 + 8.049816 + 0.387004 + 4.4521 + 5.414416 \]

Calculating this step by step:

  • \( 6.679124 + 8.049816 = 14.72894 \)
  • \( 14.72894 + 0.387004 = 15.115944 \)
  • \( 15.115944 + 4.4521 = 19.568044 \)
  • \( 19.568044 + 5.414416 = 24.98246 \) (as before)

Finally, obtain the standard deviation:

\[ \sigma = \sqrt{24.98246} \approx 4.998 \]

Clarification on the calculations

Let's ensure all computations are precise; sometimes rounding can result in slight variations:

Using a calculator for the variance:

  • \( 6.679124 + 8.049816 + 0.387004 + 4.4521 + 5.414416 \) should yield the same result as calculated stepwise.
  • \[ \sigma^2 \approx 24.98246 \Rightarrow \sqrt{24.98246} \approx 4.998 \]

If the digits in the combined variance do not throw a different standard deviation result, then rounding and precision in calculations can also change the outcome slightly.

However, based on the mathematics, the calculations above assert the standard deviation is indeed approximately 4.998 given these provided values. If there’s an exact expected deviation value needing another scrutiny, please indicate if there’s a specific area or aspect you’re noticing discrepancies.

the answer is 4.993

Thank you for your patience. Let’s carefully re-evaluate each calculation step for the variance and ultimately the standard deviation to ensure correctness.

Given Values:

  • \( x = -9, p(x) = 0.11 \)
  • \( x = -7, p(x) = 0.24 \)
  • \( x = 0, p(x) = 0.26 \)
  • \( x = 3, p(x) = 0.25 \)
  • \( x = 5, p(x) = 0.14 \)

Mean Calculation Recap

We previously calculated the mean correctly as: \[ \mu \approx -1.22 \]

Step-by-Step Variance Calculation

We calculate \( (x - \mu)^2 \cdot p(x) \) for each value of \( x \):

  1. For \( x = -9 \): \[ (-9 - (-1.22))^2 = (-7.78)^2 = 60.7284 \] Contribution: \[ 60.7284 \cdot 0.11 = 6.679124 \]

  2. For \( x = -7 \): \[ (-7 - (-1.22))^2 = (-5.78)^2 = 33.4084 \] Contribution: \[ 33.4084 \cdot 0.24 = 8.049816 \]

  3. For \( x = 0 \): \[ (0 - (-1.22))^2 = (1.22)^2 = 1.4884 \] Contribution: \[ 1.4884 \cdot 0.26 = 0.387004 \]

  4. For \( x = 3 \): \[ (3 - (-1.22))^2 = (4.22)^2 = 17.8084 \] Contribution: \[ 17.8084 \cdot 0.25 = 4.4521 \]

  5. For \( x = 5 \): \[ (5 - (-1.22))^2 = (6.22)^2 = 38.6884 \] Contribution: \[ 38.6884 \cdot 0.14 = 5.414416 \]

Summing Contributions to Calculate Variance

Now, let's sum these contributions accurately:

\[ \sigma^2 = 6.679124 + 8.049816 + 0.387004 + 4.4521 + 5.414416 \]

Calculating the total step-by-step:

  • Step 1: \( 6.679124 + 8.049816 = 14.72894 \)
  • Step 2: \( 14.72894 + 0.387004 = 15.115944 \)
  • Step 3: \( 15.115944 + 4.4521 = 19.568044 \)
  • Step 4: \( 19.568044 + 5.414416 = 24.98246 \)

Final Calculation of Standard Deviation

Now, take the square root to find \( \sigma \):

\[ \sigma = \sqrt{24.98246} \]

Calculating the square root:

  1. \( \sqrt{24.98246} \approx 4.998 \)

Checking Precision

Let’s carry out a more precise calculation of the individual contributions and the sum just to check if there were any minor discrepancies.

  1. For \( x = -9 \): \[ 60.7284 \cdot 0.11 = 6.679124 \]

  2. For \( x = -7 \): \[ 33.4084 \cdot 0.24 = 8.049816 \]

  3. For \( x = 0 \): \[ 1.4884 \cdot 0.26 = 0.387004 \]

  4. For \( x = 3 \): \[ 17.8084 \cdot 0.25 = 4.4521 \]

  5. For \( x = 5 \): \[ 38.6884 \cdot 0.14 = 5.414416 \]

Summing these again gives:

  1. \[ 6.679124 + 8.049816 = 14.72894 \]
  2. \[ 14.72894 + 0.387004 = 15.115944 \]
  3. \[ 15.115944 + 4.4521 = 19.568044 \]
  4. \[ 19.568044 + 5.414416 = 24.98246 \]

Conclusion

The computed variance is \( 24.98246 \) and upon taking the square root, we consistently find:

\[ \sigma \approx 4.998 \]

If any different calculation method or inputs led you to \( 4.993 \), do let me know, as small differences can arise due to rounding or approximations in intermediate steps. Thank you again for your attention to detail!