document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163974>Question 1163974</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The random variables (X,Y) have joint pdf f(x,y) = 1/32 for the region 0 < x < 8, 0 < y < 4. The pdf is zero outside this region. <BR>\n" );
document.write( "i. What are the marginal distributions of X and Y?<BR>\n" );
document.write( "ii. Find the joint cdf of (X,Y) F(x,y).<BR>\n" );
document.write( "iii. Find the probability P(X+Y < 4).<BR>\n" );
document.write( "iv. Find the conditional probability density f(x|y)<BR>\n" );
document.write( "v. Find the mean of X given Y i.e. E(X|Y).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854359 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=854359\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=1<BR>\n" );
document.write( "# Let's perform some basic checks for the probability and expectations<BR>\n" );
document.write( "# f(x,y) = 1/32 for 0<x<8, 0<y<4.\r<BR>\n" );
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document.write( "# iii. P(X+Y < 4)<BR>\n" );
document.write( "# Triangle with vertices (0,0), (4,0), (0,4)<BR>\n" );
document.write( "area_event = 0.5 * 4 * 4 # 8<BR>\n" );
document.write( "prob_event = area_event * (1/32) # 8/32 = 1/4\r<BR>\n" );
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document.write( "# v. E(X|Y)<BR>\n" );
document.write( "# f(x|y) = f(x,y)/f_Y(y)<BR>\n" );
document.write( "# f_Y(y) = integral_0^8 (1/32) dx = 8/32 = 1/4 for 0<y<4.<BR>\n" );
document.write( "# f(x|y) = (1/32) / (1/4) = 4/32 = 1/8 for 0<x<8.<BR>\n" );
document.write( "# E(X|Y) = integral_0^8 x * (1/8) dx = [x^2/16]_0^8 = 64/16 = 4.\r<BR>\n" );
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document.write( "print(f\"P(X+Y < 4): {prob_event}\")<BR>\n" );
document.write( "print(f\"E(X|Y): {4.0}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=1<BR>\n" );
document.write( "P(X+Y < 4): 0.25<BR>\n" );
document.write( "E(X|Y): 4.0\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "This is a uniform distribution over a rectangular region. Let's solve each part step-by-step.\r<BR>\n" );
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document.write( "### **i. What are the marginal distributions of $X$ and $Y$?**<BR>\n" );
document.write( "To find the marginals, we integrate the joint pdf over the range of the other variable.\r<BR>\n" );
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document.write( "* **For $X$ ($0 < x < 8$):**<BR>\n" );
document.write( "  $$f_X(x) = \int_0^4 \frac{1}{32} \, dy = \left[ \frac{y}{32} \right]_0^4 = \frac{4}{32} = \frac{1}{8}$$<BR>\n" );
document.write( "* **For $Y$ ($0 < y < 4$):**<BR>\n" );
document.write( "  $$f_Y(y) = \int_0^8 \frac{1}{32} \, dx = \left[ \frac{x}{32} \right]_0^8 = \frac{8}{32} = \frac{1}{4}$$\r<BR>\n" );
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document.write( "Since the marginals are constant, **$X \sim \text{Uniform}(0, 8)$** and **$Y \sim \text{Uniform}(0, 4)$**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **ii. Find the joint CDF $F(x,y)$**<BR>\n" );
document.write( "The joint CDF is the integral of the joint pdf from the lower bounds to $(x, y)$ within the region:<BR>\n" );
document.write( "$$F(x, y) = \int_0^y \int_0^x \frac{1}{32} \, du \, dv = \frac{xy}{32}$$<BR>\n" );
document.write( "This is valid for $0 < x < 8$ and $0 < y < 4$. (For values outside this range, the CDF is $0$ if $x,y \le 0$, and $1$ if $x \ge 8, y \ge 4$).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **iii. Find the probability $P(X+Y < 4)$**<BR>\n" );
document.write( "Since the distribution is uniform, the probability is simply the **area of the event region** divided by the **total area** ($8 \times 4 = 32$).<BR>\n" );
document.write( "* The region $X+Y < 4$ forms a triangle with vertices at $(0,0)$, $(4,0)$, and $(0,4)$.<BR>\n" );
document.write( "* $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8$.<BR>\n" );
document.write( "* **$P(X+Y < 4) = \frac{8}{32} = \frac{1}{4}$ (or $0.25$)**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **iv. Find the conditional probability density $f(x \mid y)$**<BR>\n" );
document.write( "The conditional density is defined as the joint density divided by the marginal density of the conditioning variable:<BR>\n" );
document.write( "$$f(x \mid y) = \frac{f(x, y)}{f_Y(y)} = \frac{1/32}{1/4} = \frac{4}{32} = \frac{1}{8}$$<BR>\n" );
document.write( "This is valid for $0 < x < 8$. Note that $f(x \mid y)$ does not depend on $y$, which proves that **$X$ and $Y$ are independent**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### **v. Find the mean of $X$ given $Y$ ($E[X \mid Y]$)**<BR>\n" );
document.write( "Since we found $f(x \mid y) = 1/8$ (a uniform distribution on $[0, 8]$):<BR>\n" );
document.write( "$$E[X \mid Y] = \int_0^8 x \cdot f(x \mid y) \, dx = \int_0^8 \frac{x}{8} \, dx$$<BR>\n" );
document.write( "$$E[X \mid Y] = \left[ \frac{x^2}{16} \right]_0^8 = \frac{64}{16} = 4$$\r<BR>\n" );
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document.write( "Because $X$ and $Y$ are independent, $E[X \mid Y]$ is simply the same as the marginal mean $E[X]$, which is the midpoint of its range $(0+8)/2 = 4$. </SPAN>\n" );
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