document.write( "Question 1202002: Karlson sends a message to Mažylis consisting of two bits in length. When he is in a good mood (with probability q), he sends '11', otherwise he sends '01'. Each bit is transmitted over the communication channel with a probability of distortion p. Let X be the number of ones in the message sent by Karlson and Y be the number of ones in the message received by Mažylis. Find the covariance between X and Y.
\n" ); document.write( "(p = 0.26, q = 0.58) \n" ); document.write( "
Algebra.Com's Answer #848450 by ElectricPavlov(122)\"\" \"About 
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**1. Determine the Probability Mass Function (PMF) of X**\r
\n" ); document.write( "\n" ); document.write( "* **Good Mood (Probability q = 0.58):** Sends \"11\"
\n" ); document.write( " * P(X=2) = 0.58
\n" ); document.write( "* **Bad Mood (Probability 1-q = 0.42):** Sends \"01\"
\n" ); document.write( " * P(X=1) = 0.42\r
\n" ); document.write( "\n" ); document.write( "**2. Determine the Possible Received Messages (Y) and their Probabilities**\r
\n" ); document.write( "\n" ); document.write( "* **\"11\" Sent:**
\n" ); document.write( " * Received \"11\": Probability = 0.58 * (1-p) * (1-p) = 0.58 * 0.74 * 0.74 = 0.3175
\n" ); document.write( " * Received \"10\": Probability = 0.58 * (1-p) * p = 0.58 * 0.74 * 0.26 = 0.1129
\n" ); document.write( " * Received \"01\": Probability = 0.58 * p * (1-p) = 0.58 * 0.26 * 0.74 = 0.1129
\n" ); document.write( " * Received \"00\": Probability = 0.58 * p * p = 0.58 * 0.26 * 0.26 = 0.0392\r
\n" ); document.write( "\n" ); document.write( "* **\"01\" Sent:**
\n" ); document.write( " * Received \"11\": Probability = 0.42 * p * (1-p) = 0.42 * 0.26 * 0.74 = 0.0808
\n" ); document.write( " * Received \"10\": Probability = 0.42 * p * p = 0.42 * 0.26 * 0.26 = 0.0283
\n" ); document.write( " * Received \"01\": Probability = 0.42 * (1-p) * p = 0.42 * 0.74 * 0.26 = 0.0808
\n" ); document.write( " * Received \"00\": Probability = 0.42 * (1-p) * (1-p) = 0.42 * 0.74 * 0.74 = 0.2289\r
\n" ); document.write( "\n" ); document.write( "**3. Calculate E[X], E[Y], and E[XY]**\r
\n" ); document.write( "\n" ); document.write( "* **E[X] (Expected value of X):**
\n" ); document.write( " * E[X] = (2 * 0.58) + (1 * 0.42) = 1.58\r
\n" ); document.write( "\n" ); document.write( "* **E[Y]:**
\n" ); document.write( " * E[Y] = (2 * 0.3175) + (1 * 0.1129 + 0.1129 + 0.0808 + 0.0808) + (0 * 0.0392 + 0.0283 + 0.2289)
\n" ); document.write( " * E[Y] = 0.635 + 0.3874 + 0.2572 = 1.2796\r
\n" ); document.write( "\n" ); document.write( "* **E[XY]:**
\n" ); document.write( " * E[XY] = (2 * 2 * 0.3175) + (2 * 1 * 0.1129) + (1 * 2 * 0.0808) + (1 * 1 * 0.0808) + (0 * 2 * 0.0392) + (0 * 1 * 0.0283) + (0 * 1 * 0.2289)
\n" ); document.write( " * E[XY] = 1.27 + 0.2258 + 0.1616 + 0.0808
\n" ); document.write( " * E[XY] = 1.7382\r
\n" ); document.write( "\n" ); document.write( "**4. Calculate Covariance**\r
\n" ); document.write( "\n" ); document.write( "* **Cov(X, Y) = E[XY] - E[X] * E[Y]**
\n" ); document.write( "* Cov(X, Y) = 1.7382 - (1.58 * 1.2796)
\n" ); document.write( "* Cov(X, Y) = 1.7382 - 2.0215
\n" ); document.write( "* Cov(X, Y) = -0.2833\r
\n" ); document.write( "\n" ); document.write( "**Therefore, the covariance between X (number of ones sent) and Y (number of ones received) is -0.2833.**\r
\n" ); document.write( "\n" ); document.write( "This negative covariance indicates that there is a tendency for the number of ones in the sent message to be inversely related to the number of ones in the received message.
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