document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162599>Question 1162599</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box contains 5 yellow balls and 3 green balls. Draw 2 balls without<BR>\n" );
document.write( "replacement. If X represents the number of yellow balls and Y represents the<BR>\n" );
document.write( "number of green balls drawn. What is the covariance of X and Y? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854400 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=854400\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "from fractions import Fraction\r<BR>\n" );
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document.write( "# Setup<BR>\n" );
document.write( "total_balls = 8<BR>\n" );
document.write( "yellow_balls = 5<BR>\n" );
document.write( "green_balls = 3<BR>\n" );
document.write( "draw_size = 2\r<BR>\n" );
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document.write( "# X is number of yellow balls<BR>\n" );
document.write( "# P(X=k) = (comb(5, k) * comb(3, 2-k)) / comb(8, 2)\r<BR>\n" );
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document.write( "def combinations(n, k):<BR>\n" );
document.write( "    if k < 0 or k > n:<BR>\n" );
document.write( "        return 0<BR>\n" );
document.write( "    import math<BR>\n" );
document.write( "    return math.comb(n, k)\r<BR>\n" );
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document.write( "denom = combinations(total_balls, draw_size)\r<BR>\n" );
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document.write( "# Probabilities<BR>\n" );
document.write( "p_x0 = Fraction(combinations(5, 0) * combinations(3, 2), denom)<BR>\n" );
document.write( "p_x1 = Fraction(combinations(5, 1) * combinations(3, 1), denom)<BR>\n" );
document.write( "p_x2 = Fraction(combinations(5, 2) * combinations(3, 0), denom)\r<BR>\n" );
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document.write( "# E[X]<BR>\n" );
document.write( "ex = 0 * p_x0 + 1 * p_x1 + 2 * p_x2\r<BR>\n" );
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document.write( "# E[X^2]<BR>\n" );
document.write( "ex2 = (0**2) * p_x0 + (1**2) * p_x1 + (2**2) * p_x2\r<BR>\n" );
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document.write( "# Var(X)<BR>\n" );
document.write( "var_x = ex2 - (ex**2)\r<BR>\n" );
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document.write( "# Cov(X, Y) = -Var(X)<BR>\n" );
document.write( "cov_xy = -var_x\r<BR>\n" );
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document.write( "print(f\"{p_x0=}\")<BR>\n" );
document.write( "print(f\"{p_x1=}\")<BR>\n" );
document.write( "print(f\"{p_x2=}\")<BR>\n" );
document.write( "print(f\"{ex=}\")<BR>\n" );
document.write( "print(f\"{ex2=}\")<BR>\n" );
document.write( "print(f\"{var_x=}\")<BR>\n" );
document.write( "print(f\"{cov_xy=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "p_x0=Fraction(3, 28)<BR>\n" );
document.write( "p_x1=Fraction(15, 28)<BR>\n" );
document.write( "p_x2=Fraction(5, 14)<BR>\n" );
document.write( "ex=Fraction(5, 4)<BR>\n" );
document.write( "ex2=Fraction(55, 28)<BR>\n" );
document.write( "var_x=Fraction(45, 112)<BR>\n" );
document.write( "cov_xy=Fraction(-45, 112)\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the covariance of $X$ (the number of yellow balls) and $Y$ (the number of green balls), we can use the properties of random variables and the specific relationship between $X$ and $Y$ in this experiment.\r<BR>\n" );
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document.write( "### 1. The Relationship between $X$ and $Y$<BR>\n" );
document.write( "Since we are drawing a total of $2$ balls and every ball drawn must be either yellow or green, we have the following constraint:<BR>\n" );
document.write( "$$X + Y = 2$$<BR>\n" );
document.write( "This implies that $Y$ is a linear function of $X$:<BR>\n" );
document.write( "$$Y = 2 - X$$\r<BR>\n" );
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document.write( "A key property of covariance is that $Cov(X, a + bX) = b \cdot Var(X)$. In this case, $a = 2$ and $b = -1$, so:<BR>\n" );
document.write( "$$Cov(X, Y) = Cov(X, 2 - X) = -Var(X)$$\r<BR>\n" );
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document.write( "### 2. Find the Variance of $X$<BR>\n" );
document.write( "$X$ follows a **hypergeometric distribution** because we are sampling without replacement from a finite population.<BR>\n" );
document.write( "* Total balls ($N$) = $8$<BR>\n" );
document.write( "* Yellow balls ($K$) = $5$<BR>\n" );
document.write( "* Number of balls drawn ($n$) = $2$\r<BR>\n" );
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document.write( "The variance of a hypergeometric distribution is given by the formula:<BR>\n" );
document.write( "$$Var(X) = n \cdot \frac{K}{N} \cdot \left( \frac{N-K}{N} \right) \cdot \left( \frac{N-n}{N-1} \right)$$\r<BR>\n" );
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document.write( "Plugging in the values:<BR>\n" );
document.write( "$$Var(X) = 2 \cdot \frac{5}{8} \cdot \frac{3}{8} \cdot \frac{8-2}{8-1}$$<BR>\n" );
document.write( "$$Var(X) = 2 \cdot \frac{5}{8} \cdot \frac{3}{8} \cdot \frac{6}{7}$$<BR>\n" );
document.write( "$$Var(X) = \frac{180}{448}$$\r<BR>\n" );
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document.write( "Simplifying the fraction:<BR>\n" );
document.write( "$$Var(X) = \frac{45}{112}$$\r<BR>\n" );
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document.write( "### 3. Calculate the Covariance<BR>\n" );
document.write( "As established in Step 1, $Cov(X, Y) = -Var(X)$.<BR>\n" );
document.write( "$$Cov(X, Y) = -\frac{45}{112}$$\r<BR>\n" );
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document.write( "### Summary of Probability Distribution (Optional Verification)<BR>\n" );
document.write( "We can also calculate $E[XY] - E[X]E[Y]$ directly:<BR>\n" );
document.write( "* **$P(X=0, Y=2)$**: $\frac{\binom{5}{0}\binom{3}{2}}{\binom{8}{2}} = \frac{3}{28}$<BR>\n" );
document.write( "* **$P(X=1, Y=1)$**: $\frac{\binom{5}{1}\binom{3}{1}}{\binom{8}{2}} = \frac{15}{28}$<BR>\n" );
document.write( "* **$P(X=2, Y=0)$**: $\frac{\binom{5}{2}\binom{3}{0}}{\binom{8}{2}} = \frac{10}{28} = \frac{5}{14}$\r<BR>\n" );
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document.write( "**Expectations:**<BR>\n" );
document.write( "* $E[X] = (0 \cdot \frac{3}{28}) + (1 \cdot \frac{15}{28}) + (2 \cdot \frac{10}{28}) = \frac{35}{28} = 1.25$<BR>\n" );
document.write( "* $E[Y] = (2 \cdot \frac{3}{28}) + (1 \cdot \frac{15}{28}) + (0 \cdot \frac{10}{28}) = \frac{21}{28} = 0.75$<BR>\n" );
document.write( "* $E[XY] = (1 \times 1) \cdot P(X=1, Y=1) = \frac{15}{28} \approx 0.5357$\r<BR>\n" );
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document.write( "**Covariance Calculation:**<BR>\n" );
document.write( "$$Cov(X, Y) = E[XY] - E[X]E[Y] = \frac{15}{28} - (1.25 \times 0.75) = \frac{15}{28} - 0.9375 = \frac{15}{28} - \frac{15}{16}$$<BR>\n" );
document.write( "$$Cov(X, Y) = \frac{60 - 105}{112} = -\frac{45}{112}$$\r<BR>\n" );
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document.write( "**Final Answer:**<BR>\n" );
document.write( "The covariance of $X$ and $Y$ is **$-\frac{45}{112}$** (or approximately **$-0.4018$**). </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );