document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204126>Question 1204126</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Let X and Y be jointly normal random variables with parameters µ_X = 1, (σ_X)^2= 1, µ_Y = 0, (σ_Y)^2= 4, and ρ = 1/2.<BR>\n" );
document.write( "(a) Find P(2X + Y < 3).<BR>\n" );
document.write( "(b) Find P(Y > 1|X = 2).<BR>\n" );
document.write( "(c) Find conditional expectation of Y given X = 2.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847992 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=847992\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Find P(2X + Y < 3)**\r<BR>\n" );
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document.write( "* **Define a new random variable:** Let Z = 2X + Y \r<BR>\n" );
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document.write( "* **Find the mean and variance of Z:**\r<BR>\n" );
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document.write( "    * E[Z] = E[2X + Y] = 2E[X] + E[Y] = 2(1) + 0 = 2<BR>\n" );
document.write( "    * Var(Z) = Var(2X + Y) = 4Var(X) + Var(Y) + 2 * 2 * Cov(X, Y) <BR>\n" );
document.write( "      * Cov(X, Y) = ρ * σ_X * σ_Y = (1/2) * 1 * 2 = 1<BR>\n" );
document.write( "    * Var(Z) = 4 * 1 + 4 + 2 * 1 = 10\r<BR>\n" );
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document.write( "* **Determine the distribution of Z:**\r<BR>\n" );
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document.write( "    * Since X and Y are jointly normal, any linear combination of them (like Z) is also normally distributed.\r<BR>\n" );
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document.write( "* **Standardize Z:**\r<BR>\n" );
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document.write( "    * Let W = (Z - E[Z]) / sqrt(Var(Z)) <BR>\n" );
document.write( "      * W = (Z - 2) / sqrt(10) <BR>\n" );
document.write( "      * W follows a standard normal distribution (N(0, 1))\r<BR>\n" );
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document.write( "* **Calculate the probability:**\r<BR>\n" );
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document.write( "    * P(2X + Y < 3) = P(Z < 3) <BR>\n" );
document.write( "      * P(Z < 3) = P( (Z - 2) / sqrt(10) < (3 - 2) / sqrt(10) ) <BR>\n" );
document.write( "      * P(Z < 3) = P(W < 1 / sqrt(10)) <BR>\n" );
document.write( "      * Use a standard normal distribution table or software to find P(W < 1 / sqrt(10)) \r<BR>\n" );
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document.write( "**b) Find P(Y > 1 | X = 2)**\r<BR>\n" );
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document.write( "* **Conditional Distribution of Y given X:**\r<BR>\n" );
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document.write( "    * When X and Y are jointly normal, the conditional distribution of Y given X is also normal.<BR>\n" );
document.write( "    * The conditional mean of Y given X is:<BR>\n" );
document.write( "        * E[Y | X] = μ_Y + ρ * (σ_Y / σ_X) * (X - μ_X) <BR>\n" );
document.write( "          * E[Y | X = 2] = 0 + (1/2) * (2 / 1) * (2 - 1) = 1\r<BR>\n" );
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document.write( "    * The conditional variance of Y given X is:<BR>\n" );
document.write( "        * Var(Y | X) = σ_Y² * (1 - ρ²) <BR>\n" );
document.write( "          * Var(Y | X) = 4 * (1 - (1/2)²) = 4 * (3/4) = 3\r<BR>\n" );
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document.write( "* **Calculate the probability:**\r<BR>\n" );
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document.write( "    * P(Y > 1 | X = 2) = P( (Y - E[Y | X = 2]) / sqrt(Var(Y | X)) > (1 - 1) / sqrt(3) ) <BR>\n" );
document.write( "      * P(Y > 1 | X = 2) = P(Z > 0) <BR>\n" );
document.write( "        * where Z is a standard normal random variable. <BR>\n" );
document.write( "      * P(Y > 1 | X = 2) = 0.5 \r<BR>\n" );
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document.write( "**c) Find the conditional expectation of Y given X = 2**\r<BR>\n" );
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document.write( "* As calculated in part (b):<BR>\n" );
document.write( "    * E[Y | X = 2] = 1\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* **(a) P(2X + Y < 3)** requires standardizing the linear combination of X and Y and then using a standard normal distribution table.<BR>\n" );
document.write( "* **(b) P(Y > 1 | X = 2)** utilizes the properties of the conditional distribution of Y given X in a jointly normal distribution.<BR>\n" );
document.write( "* **(c) The conditional expectation of Y given X = 2** is calculated directly using the formula for the conditional mean.\r<BR>\n" );
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document.write( "I hope this comprehensive explanation is helpful!<BR>\n" );
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