document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1207013>Question 1207013</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4. Discrete independent random variables X and Y are given by the following laws of distribution:<BR>\n" );
document.write( "X 0 3 Y - 2 - 1 2<BR>\n" );
document.write( "P 0,3 0,7 P 0,2 0,4 0,4<BR>\n" );
document.write( "Find M (X + Y) by two ways: 1) composing the law of distribution of X + Y; 2) using the property: M (X + Y) = M (X) + M (Y). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847996 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847996\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> We are tasked with calculating the expected value \( M(X + Y) \) by two methods. Let us proceed step by step.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 1: Distribution Laws**<BR>\n" );
document.write( "#### Distribution of \( X \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "X: \quad 0, \, 3 \quad \text{with probabilities } \quad P(X = 0) = 0.3, \, P(X = 3) = 0.7.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Distribution of \( Y \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "Y: \quad -2, \, -1, \, 2 \quad \text{with probabilities } \quad P(Y = -2) = 0.2, \, P(Y = -1) = 0.4, \, P(Y = 2) = 0.4.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "### **Step 2: Method 1 - Composing the Law of Distribution for \( X + Y \)**<BR>\n" );
document.write( "Since \( X \) and \( Y \) are independent, the probability of any combination \( (X, Y) \) is given by the product of their probabilities:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X + Y = z) = \sum_{(x, y): x + y = z} P(X = x) \cdot P(Y = y).<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Compute Possible Values of \( X + Y \):<BR>\n" );
document.write( "- If \( X = 0 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  X + Y \in \{-2, -1, 2\} \quad \text{(corresponding to \( Y = -2, -1, 2 \))}.<BR>\n" );
document.write( "  \]<BR>\n" );
document.write( "- If \( X = 3 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  X + Y \in \{1, 2, 5\} \quad \text{(corresponding to \( Y = -2, -1, 2 \))}.<BR>\n" );
document.write( "  \]\r<BR>\n" );
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document.write( "Thus, the possible values of \( X + Y \) are:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "\{-2, -1, 1, 2, 5\}.<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "#### Compute Probabilities for Each Value of \( X + Y \):<BR>\n" );
document.write( "- For \( X + Y = -2 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  P(X + Y = -2) = P(X = 0) \cdot P(Y = -2) = 0.3 \cdot 0.2 = 0.06.<BR>\n" );
document.write( "  \]<BR>\n" );
document.write( "- For \( X + Y = -1 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  P(X + Y = -1) = P(X = 0) \cdot P(Y = -1) = 0.3 \cdot 0.4 = 0.12.<BR>\n" );
document.write( "  \]<BR>\n" );
document.write( "- For \( X + Y = 1 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  P(X + Y = 1) = P(X = 3) \cdot P(Y = -2) = 0.7 \cdot 0.2 = 0.14.<BR>\n" );
document.write( "  \]<BR>\n" );
document.write( "- For \( X + Y = 2 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  P(X + Y = 2) = P(X = 0) \cdot P(Y = 2) + P(X = 3) \cdot P(Y = -1) = (0.3 \cdot 0.4) + (0.7 \cdot 0.4) = 0.12 + 0.28 = 0.4.<BR>\n" );
document.write( "  \]<BR>\n" );
document.write( "- For \( X + Y = 5 \):<BR>\n" );
document.write( "  \[<BR>\n" );
document.write( "  P(X + Y = 5) = P(X = 3) \cdot P(Y = 2) = 0.7 \cdot 0.4 = 0.28.<BR>\n" );
document.write( "  \]\r<BR>\n" );
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document.write( "#### Distribution of \( X + Y \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "X + Y: \quad -2, \, -1, \, 1, \, 2, \, 5 \quad \text{with probabilities } \quad 0.06, \, 0.12, \, 0.14, \, 0.4, \, 0.28.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Compute \( M(X + Y) \):<BR>\n" );
document.write( "The expected value is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X + Y) = \sum_{z} z \cdot P(X + Y = z).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Substitute:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X + Y) = (-2)(0.06) + (-1)(0.12) + (1)(0.14) + (2)(0.4) + (5)(0.28).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X + Y) = -0.12 - 0.12 + 0.14 + 0.8 + 1.4 = 2.1.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Step 3: Method 2 - Using the Property \( M(X + Y) = M(X) + M(Y) \)**<BR>\n" );
document.write( "#### Compute \( M(X) \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X) = \sum_{x} x \cdot P(X = x).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X) = (0)(0.3) + (3)(0.7) = 2.1.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "#### Compute \( M(Y) \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(Y) = \sum_{y} y \cdot P(Y = y).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(Y) = (-2)(0.2) + (-1)(0.4) + (2)(0.4).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(Y) = -0.4 - 0.4 + 0.8 = 0.<BR>\n" );
document.write( "\]\r<BR>\n" );
document.write( "\n" );
document.write( "#### Compute \( M(X + Y) \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X + Y) = M(X) + M(Y) = 2.1 + 0 = 2.1.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **Final Answer**:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "M(X + Y) = \boxed{2.1}.<BR>\n" );
document.write( "\] </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );