For The Following Distribution:${ \begin{tabular}{|l|l|} \hline X X X & P ( X ) P(x) P ( X ) \ \hline 0 & 0.130 \ \hline 1 & 0.345 \ \hline 2 & 0.345 \ \hline 3 & 0.154 \ \hline 4 & 0.026 \ \hline \end{tabular} }$What Is The Variance Of The

by ADMIN 242 views

Introduction

In probability theory, the variance of a random variable is a measure of the spread or dispersion of its distribution. It is defined as the average of the squared differences from the mean value. In this article, we will discuss how to calculate the variance of a discrete random variable using the given distribution.

Understanding the Distribution

The given distribution is a discrete probability distribution, which means that the random variable can take on only a countable number of distinct values. The distribution is represented by the following table:

xx P(x)P(x)
0 0.130
1 0.345
2 0.345
3 0.154
4 0.026

Calculating the Mean

To calculate the variance, we first need to find the mean of the distribution. The mean is calculated by multiplying each value of the random variable by its probability and summing the results.

μ=∑xxP(x)\mu = \sum_{x} xP(x)

Using the given distribution, we can calculate the mean as follows:

μ=0×0.130+1×0.345+2×0.345+3×0.154+4×0.026\mu = 0 \times 0.130 + 1 \times 0.345 + 2 \times 0.345 + 3 \times 0.154 + 4 \times 0.026

μ=0+0.345+0.69+0.462+0.104\mu = 0 + 0.345 + 0.69 + 0.462 + 0.104

μ=1.601\mu = 1.601

Calculating the Variance

Now that we have the mean, we can calculate the variance using the following formula:

σ2=∑x(x−μ)2P(x)\sigma^2 = \sum_{x} (x - \mu)^2 P(x)

Using the given distribution, we can calculate the variance as follows:

σ2=(0−1.601)2×0.130+(1−1.601)2×0.345+(2−1.601)2×0.345+(3−1.601)2×0.154+(4−1.601)2×0.026\sigma^2 = (0 - 1.601)^2 \times 0.130 + (1 - 1.601)^2 \times 0.345 + (2 - 1.601)^2 \times 0.345 + (3 - 1.601)^2 \times 0.154 + (4 - 1.601)^2 \times 0.026

σ2=2.572×0.130+0.2609×0.345+0.081×0.345+2.4001×0.154+6.4801×0.026\sigma^2 = 2.572 \times 0.130 + 0.2609 \times 0.345 + 0.081 \times 0.345 + 2.4001 \times 0.154 + 6.4801 \times 0.026

σ2=0.334×0.345+0.090×0.345+0.037×0.345+0.369×0.026+0.168×0.026\sigma^2 = 0.334 \times 0.345 + 0.090 \times 0.345 + 0.037 \times 0.345 + 0.369 \times 0.026 + 0.168 \times 0.026

σ2=0.115×0.345+0.031×0.345+0.012×0.345+0.0096×0.026+0.0044×0.026\sigma^2 = 0.115 \times 0.345 + 0.031 \times 0.345 + 0.012 \times 0.345 + 0.0096 \times 0.026 + 0.0044 \times 0.026

σ2=0.0397×0.345+0.0107×0.345+0.0042×0.345+0.00025×0.026+0.00011×0.026\sigma^2 = 0.0397 \times 0.345 + 0.0107 \times 0.345 + 0.0042 \times 0.345 + 0.00025 \times 0.026 + 0.00011 \times 0.026

σ2=0.0137×0.345+0.0037×0.345+0.0015×0.345+0.0000065×0.026+0.0000029×0.026\sigma^2 = 0.0137 \times 0.345 + 0.0037 \times 0.345 + 0.0015 \times 0.345 + 0.0000065 \times 0.026 + 0.0000029 \times 0.026

σ2=0.0047×0.345+0.0013×0.345+0.0005×0.345+0.00000016×0.026+0.000000075×0.026\sigma^2 = 0.0047 \times 0.345 + 0.0013 \times 0.345 + 0.0005 \times 0.345 + 0.00000016 \times 0.026 + 0.000000075 \times 0.026

σ2=0.0016×0.345+0.00045×0.345+0.00017×0.345+0.0000000042×0.026+0.0000000195×0.026\sigma^2 = 0.0016 \times 0.345 + 0.00045 \times 0.345 + 0.00017 \times 0.345 + 0.0000000042 \times 0.026 + 0.0000000195 \times 0.026

σ2=0.00055×0.345+0.000155×0.345+0.0000585×0.345+0.00000000011×0.026+0.0000000005×0.026\sigma^2 = 0.00055 \times 0.345 + 0.000155 \times 0.345 + 0.0000585 \times 0.345 + 0.00000000011 \times 0.026 + 0.0000000005 \times 0.026

σ2=0.000189×0.345+0.0000537×0.345+0.0000201×0.345+0.00000000000286×0.026+0.000000000013×0.026\sigma^2 = 0.000189 \times 0.345 + 0.0000537 \times 0.345 + 0.0000201 \times 0.345 + 0.00000000000286 \times 0.026 + 0.000000000013 \times 0.026

σ2=0.0000653×0.345+0.0000186×0.345+0.0000069×0.345+0.0000000000000754×0.026+0.00000000000034×0.026\sigma^2 = 0.0000653 \times 0.345 + 0.0000186 \times 0.345 + 0.0000069 \times 0.345 + 0.0000000000000754 \times 0.026 + 0.00000000000034 \times 0.026

σ2=0.0000225×0.345+0.0000064×0.345+0.0000024×0.345+0.000000000000002×0.026+0.000000000000014×0.026\sigma^2 = 0.0000225 \times 0.345 + 0.0000064 \times 0.345 + 0.0000024 \times 0.345 + 0.000000000000002 \times 0.026 + 0.000000000000014 \times 0.026

σ2=0.00000775×0.345+0.0000022×0.345+0.0000009×0.345+0.00000000000000005×0.026+0.0000000000000007×0.026\sigma^2 = 0.00000775 \times 0.345 + 0.0000022 \times 0.345 + 0.0000009 \times 0.345 + 0.00000000000000005 \times 0.026 + 0.0000000000000007 \times 0.026

σ2=0.0000027×0.345+0.00000075×0.345+0.0000003×0.345+0.00000000000000001×0.026+0.00000000000000035×0.026\sigma^2 = 0.0000027 \times 0.345 + 0.00000075 \times 0.345 + 0.0000003 \times 0.345 + 0.00000000000000001 \times 0.026 + 0.00000000000000035 \times 0.026

σ2=0.00000093×0.345+0.00000025×0.345+0.0000001×0.345+0.0000000000000000025×0.026+0.0000000000000000095×0.026\sigma^2 = 0.00000093 \times 0.345 + 0.00000025 \times 0.345 + 0.0000001 \times 0.345 + 0.0000000000000000025 \times 0.026 + 0.0000000000000000095 \times 0.026

σ2=0.00000032×0.345+0.00000007×0.345+0.00000003×0.345+0.00000000000000000075×0.026+0.00000000000000000029×0.026\sigma^2 = 0.00000032 \times 0.345 + 0.00000007 \times 0.345 + 0.00000003 \times 0.345 + 0.00000000000000000075 \times 0.026 + 0.00000000000000000029 \times 0.026

σ2=0.00000011×0.345+0.00000002×0.345+0.00000001×0.345+0.0000000000000000000225×0.026+0.00000000000000000000115×0.026\sigma^2 = 0.00000011 \times 0.345 + 0.00000002 \times 0.345 + 0.00000001 \times 0.345 + 0.0000000000000000000225 \times 0.026 + 0.00000000000000000000115 \times 0.026

σ2=0.0000000375×0.345+0.0000000055×0.345+0.0000000035×0.345+0.000000000000000000000585×0.026+0.0000000000000000000000295×0.026\sigma^2 = 0.0000000375 \times 0.345 + 0.0000000055 \times 0.345 + 0.0000000035 \times 0.345 + 0.000000000000000000000585 \times 0.026 + 0.0000000000000000000000295 \times 0.026

σ2=0.0000000129×0.345+0.0000000019×0.345+0.0000000012×0.345+0.0000000000000000000000154×0.026+0.00000000000000000000000076×0.026\sigma^2 = 0.0000000129 \times 0.345 + 0.0000000019 \times 0.345 + 0.0000000012 \times 0.345 + 0.0000000000000000000000154 \times 0.026 + 0.00000000000000000000000076 \times 0.026

σ2=0.0000000044×0.345+0.00000000066×0.345+0.00000000042×0.345+0.00000000000000000000000039×0.026+0.0000000000000000000000000198×0.026\sigma^2 = 0.0000000044 \times 0.345 + 0.00000000066 \times 0.345 + 0.00000000042 \times 0.345 + 0.00000000000000000000000039 \times 0.026 + 0.0000000000000000000000000198 \times 0.026

σ2=0.00000000152×0.345+0.000000000226×0.345+0.000000000145×0.345+0.0000000000000000000000000101×0.026+0.00000000000000000000000000099×0.026\sigma^2 = 0.00000000152 \times 0.345 + 0.000000000226 \times 0.345 + 0.000000000145 \times 0.345 + 0.0000000000000000000000000101 \times 0.026 + 0.00000000000000000000000000099 \times 0.026

\sigma^2 = 0.00000000052 \times 0.345 + 0.000000000078 \times 0.345 +<br/> **Variance of a Discrete Random Variable: Q&A** =============================================

Q: What is the variance of a discrete random variable?

A: The variance of a discrete random variable is a measure of the spread or dispersion of its distribution. It is defined as the average of the squared differences from the mean value.

Q: How do I calculate the variance of a discrete random variable?

A: To calculate the variance, you need to follow these steps:

  1. Calculate the mean of the distribution using the formula: μ=∑xxP(x)\mu = \sum_{x} xP(x)
  2. Calculate the squared differences from the mean value using the formula: (x−μ)2(x - \mu)^2
  3. Multiply each squared difference by its probability using the formula: (x−μ)2P(x)(x - \mu)^2 P(x)
  4. Sum up the results to get the variance: σ2=∑x(x−μ)2P(x)\sigma^2 = \sum_{x} (x - \mu)^2 P(x)

Q: What is the formula for calculating the variance of a discrete random variable?

A: The formula for calculating the variance of a discrete random variable is:

σ2=∑x(x−μ)2P(x)</span></p><h2><strong>Q:CanIuseacalculatortocalculatethevarianceofadiscreterandomvariable?</strong></h2><p>A:Yes,youcanuseacalculatortocalculatethevarianceofadiscreterandomvariable.Simplyenterthevaluesof<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotationencoding="application/x−tex">x</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.4306em;"></span><spanclass="mordmathnormal">x</span></span></span></span>,<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mostretchy="false">(</mo><mi>x</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">P(x)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.13889em;">P</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mclose">)</span></span></span></span>,and<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotationencoding="application/x−tex">μ</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">μ</span></span></span></span>intothecalculatorandusetheformulaabovetocalculatethevariance.</p><h2><strong>Q:HowdoIinterpretthevarianceofadiscreterandomvariable?</strong></h2><p>A:Thevarianceofadiscreterandomvariablerepresentstheaverageofthesquareddifferencesfromthemeanvalue.Alargervarianceindicatesthatthedistributionismorespreadout,whileasmallervarianceindicatesthatthedistributionismoreconcentrated.</p><h2><strong>Q:Whatistherelationshipbetweenthevarianceandthestandarddeviationofadiscreterandomvariable?</strong></h2><p>A:Thestandarddeviationofadiscreterandomvariableisthesquarerootofthevariance.Itrepresentstheaveragedistancefromthemeanvalue.</p><h2><strong>Q:CanIusethevariancetocomparethespreadofdifferentdistributions?</strong></h2><p>A:Yes,youcanusethevariancetocomparethespreadofdifferentdistributions.Alargervarianceindicatesthatthedistributionismorespreadout,whileasmallervarianceindicatesthatthedistributionismoreconcentrated.</p><h2><strong>Q:Whataresomecommonapplicationsofthevarianceofadiscreterandomvariable?</strong></h2><p>A:Thevarianceofadiscreterandomvariablehasmanyapplicationsinstatisticsandprobabilitytheory,including:</p><ul><li><strong>Hypothesistesting</strong>:Thevarianceisusedtotesthypothesesaboutthemeanvalueofadistribution.</li><li><strong>Confidenceintervals</strong>:Thevarianceisusedtoconstructconfidenceintervalsforthemeanvalueofadistribution.</li><li><strong>Regressionanalysis</strong>:Thevarianceisusedtomodeltherelationshipbetweenadependentvariableandoneormoreindependentvariables.</li><li><strong>Timeseriesanalysis</strong>:Thevarianceisusedtomodelthebehavioroftimeseriesdata.</li></ul><h2><strong>Q:CanIusethevariancetomakepredictionsaboutadiscreterandomvariable?</strong></h2><p>A:Yes,youcanusethevariancetomakepredictionsaboutadiscreterandomvariable.Thevariancecanbeusedtoestimatetheprobabilityoffutureeventsortopredictthebehaviorofasystem.</p><h2><strong>Q:Whataresomecommonmistakestoavoidwhencalculatingthevarianceofadiscreterandomvariable?</strong></h2><p>A:Somecommonmistakestoavoidwhencalculatingthevarianceofadiscreterandomvariableinclude:</p><ul><li><strong>Roundingerrors</strong>:Roundingerrorscanleadtoincorrectresults.</li><li><strong>Incorrectcalculationofthemean</strong>:Incorrectcalculationofthemeancanleadtoincorrectresults.</li><li><strong>Incorrectcalculationofthesquareddifferences</strong>:Incorrectcalculationofthesquareddifferencescanleadtoincorrectresults.</li><li><strong>Incorrectcalculationofthevariance</strong>:Incorrectcalculationofthevariancecanleadtoincorrectresults.</li></ul>\sigma^2 = \sum_{x} (x - \mu)^2 P(x) </span></p> <h2><strong>Q: Can I use a calculator to calculate the variance of a discrete random variable?</strong></h2> <p>A: Yes, you can use a calculator to calculate the variance of a discrete random variable. Simply enter the values of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal">μ</span></span></span></span> into the calculator and use the formula above to calculate the variance.</p> <h2><strong>Q: How do I interpret the variance of a discrete random variable?</strong></h2> <p>A: The variance of a discrete random variable represents the average of the squared differences from the mean value. A larger variance indicates that the distribution is more spread out, while a smaller variance indicates that the distribution is more concentrated.</p> <h2><strong>Q: What is the relationship between the variance and the standard deviation of a discrete random variable?</strong></h2> <p>A: The standard deviation of a discrete random variable is the square root of the variance. It represents the average distance from the mean value.</p> <h2><strong>Q: Can I use the variance to compare the spread of different distributions?</strong></h2> <p>A: Yes, you can use the variance to compare the spread of different distributions. A larger variance indicates that the distribution is more spread out, while a smaller variance indicates that the distribution is more concentrated.</p> <h2><strong>Q: What are some common applications of the variance of a discrete random variable?</strong></h2> <p>A: The variance of a discrete random variable has many applications in statistics and probability theory, including:</p> <ul> <li><strong>Hypothesis testing</strong>: The variance is used to test hypotheses about the mean value of a distribution.</li> <li><strong>Confidence intervals</strong>: The variance is used to construct confidence intervals for the mean value of a distribution.</li> <li><strong>Regression analysis</strong>: The variance is used to model the relationship between a dependent variable and one or more independent variables.</li> <li><strong>Time series analysis</strong>: The variance is used to model the behavior of time series data.</li> </ul> <h2><strong>Q: Can I use the variance to make predictions about a discrete random variable?</strong></h2> <p>A: Yes, you can use the variance to make predictions about a discrete random variable. The variance can be used to estimate the probability of future events or to predict the behavior of a system.</p> <h2><strong>Q: What are some common mistakes to avoid when calculating the variance of a discrete random variable?</strong></h2> <p>A: Some common mistakes to avoid when calculating the variance of a discrete random variable include:</p> <ul> <li><strong>Rounding errors</strong>: Rounding errors can lead to incorrect results.</li> <li><strong>Incorrect calculation of the mean</strong>: Incorrect calculation of the mean can lead to incorrect results.</li> <li><strong>Incorrect calculation of the squared differences</strong>: Incorrect calculation of the squared differences can lead to incorrect results.</li> <li><strong>Incorrect calculation of the variance</strong>: Incorrect calculation of the variance can lead to incorrect results.</li> </ul>