An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $μ$ is the average marks of girls and $σ^{2}$ is the variance of marks of 50 candidates, then $μ + σ^{2}$ is equal to____

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alok kumar singh | Contributor-Level 10

2 months ago

Given variance of boys $σ_{b}^{2} = 2 & {\bar{x}}_{b} = 12$ (average marks of boys)

& variance of girls $σ_{g}^{2} = 2 & μ \to$ average marks of girls

$N o w, {\bar{x}}_{g} = μ = \frac{50 \times 15 - 12 \times 20}{30} = 17$

$σ^{2} = \frac{20 \times 2 + 30 \times 2}{20 + 30} + \frac{20 \times 30}{{(20 + 30)}^{2}} {(12 - 17)}^{2} = 8$

$S o, μ + σ^{2} = 17 + 8 = 25$

Given variance of boys <math> <mrow> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>b</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo>&</mo> <msub> <mrow> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> </mrow> <mrow> <mi>b</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mn>2</mn> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1815989053"> </o:OLEObject></xml><! [endif]-> (average marks of boys)& variance of girls  <math> <mrow> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>g</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <mn>2</mn> <mo>&</mo> <mi>μ</mi> <mo>→</mo> </mrow> </math> <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1026" DrawAspect="Content" ObjectID="_1815989054"> </o:OLEObject></xml><! [endif]->average marks of girls <math> <mrow> <mi>N</mi> <mi>o</mi> <mi>w</mi> <mo>, </mo> <msub> <mrow> <mover accent="true"> <mi>x</mi> <mo>¯</mo> </mover> </mrow> <mrow> <mi>g</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mo>=</mo> <mfrac> <mrow> <mn>5</mn> <mn>0</mn> <mo>×</mo> <mn>1</mn> <mn>5</mn> <mo>−</mo> <mn>1</mn> <mn>2</mn> <mo>×</mo> <mn>2</mn> <mn>0</mn> </mrow> <mrow> <mn>3</mn> <mn>0</mn> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mn>7</mn> </mrow> </math>           <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1815989055"> </o:OLEObject></xml><! [endif]-> <math> <mrow> <msup> <mrow> <mi>σ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mn>0</mn> <mo>×</mo> <mn>2</mn> <mo>+</mo> <mn>3</mn> <mn>0</mn> <mo>×</mo> <mn>2</mn> </mrow> <mrow> <mn>2</mn> <mn>0</mn> <mo>+</mo> <mn>3</mn> <mn>0</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <mn>0</mn> <mo>×</mo> <mn>3</mn> <mn>0</mn> </mrow> <mrow> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>2</mn> <mn>0</mn> <mo>+</mo> <mn>3</mn> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> <msup> <mrow> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mn>2</mn> <mo>−</mo> <mn>1</mn> <mn>7</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>8</mn> </mrow> </math>             <math> <mrow> <mi>S</mi> <mi>o</mi> <mo>, </mo> <mi>μ</mi> <mo>+</mo> <msup> <mrow> <mi>σ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mn>7</mn> <mo>+</mo> <mn>8</mn> <mo>=</mo> <mn>2</mn> <mn>5</mn> </mrow> </math>  <!- [if gte mso 9]><xml> <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1028" DrawAspect="Content" ObjectID="_1815989056"> </o:OLEObject></xml><! [endif]->

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