document.write( "Question 1193544: The random variable X has a probability distribution
\n" ); document.write( "x. 1 2 3 4 5
\n" ); document.write( "P X x ( = ) 0.1 p q 0.3 0.1
\n" ); document.write( "i. Given that E X( ) = 3.1, write down two equations involving p and q .
\n" ); document.write( "ii. Find the values of p and q .
\n" ); document.write( "iii. Find Var X( )
\n" ); document.write( "iv. Var X (2 3 . \n" ); document.write( "
Algebra.Com's Answer #848646 by yurtman(42)\"\" \"About 
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**i. Find two equations involving p and q:**\r
\n" ); document.write( "\n" ); document.write( "1. **Sum of Probabilities:**
\n" ); document.write( " The sum of all probabilities in a probability distribution must equal 1.
\n" ); document.write( " Therefore: 0.1 + p + q + 0.3 + 0.1 = 1
\n" ); document.write( " Simplifying: p + q = 0.5 \r
\n" ); document.write( "\n" ); document.write( "2. **Expected Value (E(X)):**
\n" ); document.write( " E(X) = Σ [x * P(X = x)]
\n" ); document.write( " Given E(X) = 3.1, we have:
\n" ); document.write( " 3.1 = (1 * 0.1) + (2 * p) + (3 * q) + (4 * 0.3) + (5 * 0.1)
\n" ); document.write( " 3.1 = 0.1 + 2p + 3q + 1.2 + 0.5
\n" ); document.write( " 3.1 = 1.8 + 2p + 3q
\n" ); document.write( " Simplifying: 2p + 3q = 1.3\r
\n" ); document.write( "\n" ); document.write( "**ii. Find the values of p and q:**\r
\n" ); document.write( "\n" ); document.write( "* We have two equations:
\n" ); document.write( " * p + q = 0.5
\n" ); document.write( " * 2p + 3q = 1.3\r
\n" ); document.write( "\n" ); document.write( "* Solve this system of equations:
\n" ); document.write( " * From the first equation, isolate p: p = 0.5 - q
\n" ); document.write( " * Substitute this value of p into the second equation:
\n" ); document.write( " 2(0.5 - q) + 3q = 1.3
\n" ); document.write( " 1 - 2q + 3q = 1.3
\n" ); document.write( " q = 0.3
\n" ); document.write( " * Substitute the value of q back into the first equation:
\n" ); document.write( " p + 0.3 = 0.5
\n" ); document.write( " p = 0.2\r
\n" ); document.write( "\n" ); document.write( "* Therefore, p = 0.2 and q = 0.3\r
\n" ); document.write( "\n" ); document.write( "**iii. Find Var(X)**\r
\n" ); document.write( "\n" ); document.write( "* **Calculate E(X²)**:
\n" ); document.write( " E(X²) = Σ [x² * P(X = x)]
\n" ); document.write( " E(X²) = (1² * 0.1) + (2² * 0.2) + (3² * 0.3) + (4² * 0.3) + (5² * 0.1)
\n" ); document.write( " E(X²) = 0.1 + 0.8 + 2.7 + 4.8 + 2.5
\n" ); document.write( " E(X²) = 10.9\r
\n" ); document.write( "\n" ); document.write( "* **Calculate Var(X)**:
\n" ); document.write( " Var(X) = E(X²) - [E(X)]²
\n" ); document.write( " Var(X) = 10.9 - (3.1)²
\n" ); document.write( " Var(X) = 10.9 - 9.61
\n" ); document.write( " Var(X) = 1.29\r
\n" ); document.write( "\n" ); document.write( "**iv. Find Var(2X - 3)**\r
\n" ); document.write( "\n" ); document.write( "* Use the property: Var(aX + b) = a² * Var(X)
\n" ); document.write( " where a = 2 and b = -3\r
\n" ); document.write( "\n" ); document.write( "* Var(2X - 3) = 2² * Var(X)
\n" ); document.write( " Var(2X - 3) = 4 * 1.29
\n" ); document.write( " Var(2X - 3) = 5.16\r
\n" ); document.write( "\n" ); document.write( "**Summary:**\r
\n" ); document.write( "\n" ); document.write( "* p = 0.2
\n" ); document.write( "* q = 0.3
\n" ); document.write( "* Var(X) = 1.29
\n" ); document.write( "* Var(2X - 3) = 5.16
\n" ); document.write( " \n" ); document.write( "
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