Question

From a box of fruit containing 62 oranges and 1 apple a random sample of 2 pieces of fruit has been selected without replacement. Let X be the number of oranges and Y be the number of apples in the sample. What will the correlation of X and Y be?

(Hint the variance of X and Y same.

Covariance is just negative of these variances. )

Answer #1

from above:

x | y | f(x,y) | x*f(x,y) | y*f(x,y) | x^2f(x,y) | y^2f(x,y) | xy*f(x,y) |

1 | 1 | 2/63 | 0.0317 | 0.0317 | 0.0317 | 0.0317 | 0.0317 |

2 | 0 | 61/63 | 1.9365 | 0.0000 | 3.8730 | 0.0000 | 0.0000 |

Total | 1 | 1.9683 | 0.0317 | 3.9048 | 0.0317 | 0.0317 | |

E(X)=ΣxP(x,y)= | 1.9683 | ||||||

E(X^{2})=Σx^{2}P(x,y)= |
3.9048 | ||||||

E(Y)=ΣyP(x,y)= | 0.0317 | ||||||

E(Y^{2})=Σy^{2}P(x,y)= |
0.0317 | ||||||

Var(X)=E(X^{2})-(E(X))^{2}= |
0.0307 | ||||||

Var(Y)=E(Y^{2})-(E(Y))^{2}= |
0.0307 | ||||||

E(XY)=ΣxyP(x,y)= | 0.0317 | ||||||

Cov(X,Y)=E(XY)-E(X)*E(Y)= | -0.0307 | ||||||

Correlation ρxy=Cov(X,Y)/sqrt(Var(X)*Var(Y))= |
-1.0000 |

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