Suppose we draw 5 cards from a standard deck of cards withou

Suppose we draw 5 cards from a standard deck of cards without replacement. Let X denote the numberof spades in our hand and let Y denote the number of jacks in our hand. Find:a) E(X), E(Y )b) V ar(X), V ar(Y )c) Cov(X, Y ), Corr(X, Y ).

Solution

Probability distribution for X

P (X=0) = ³⁹ C ₅ / ⁵² C ₅ = 2109 / 9520

P(X = 1) = ³⁹C₄¹³C₁ / ⁵² C ₅ = 0.4114

P(X = 2) = ³⁹C₃¹³C₂ / ⁵² C ₅ = 0.2742

P(X = 3) = ³⁹C₂¹³C₃ / ⁵² C ₅ = 0.08154

P(X = 4) = ³⁹C₁¹³C₄ / ⁵² C ₅ = 0.01072

P(X = 5) = ³⁹C₀¹³C₅ / ⁵² C ₅ = 0.00049

Thus, E(X) = 1(0.4114) + 2(0.2742) + 3(0.08154) + 4(0.01072) +5(0.00049) =1.24975 ~ 1.25

Var (X) = [1²(0.4114) + 2²(0.2742) + 3²(0.08154) + 4²(0.01072) +5²(0.00049)] - (1.25)²

= 0.86333

Now,

P(Y=0) = ⁴⁸C₅⁴C₀ / ⁵² C ₅ = 0.6588

P(Y=1) = ⁴⁸C₄⁴C₁ / ⁵² C ₅ = 0.29947

P(Y=2) = ⁴⁸C₃⁴C₂ / ⁵² C ₅ = 0.0399

P(Y=3) = ⁴⁸C₂⁴C₃ / ⁵² C ₅ = 0.00173

P(Y=4) = ⁴⁸C₁⁴C₄ / ⁵² C ₅ = 1.84 * 10^-5

Thus,

E(Y) = 1(0.29947) + 2(0.0399) + 3(0.00173) + 4(1.84 * 10^-5) =0.3845

Var (Y) = [1²(0.29947) + 2²(0.0399) + 3²(0.00173) + 4²(1.84 * 10^-5)] - (0.3845)²

= 0.3270

Now,

Cov (X,Y) = E(XY) - E(X)E(Y)

(We need a joint probability distribution table for this one) Knowing the probability will make it easier to calculate.
Correlation is given by:

_XY=Corr(X,Y)=Cov(X,Y)/ _XY

Hope this helps. Ask if you have further queries.