document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1164365>Question 1164365</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. \r<BR>\n" );
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document.write( "iii. Find the probability P(X+Y < 4).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854352 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=854352\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import scipy.integrate as integrate\r<BR>\n" );
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document.write( "# Define the pdf<BR>\n" );
document.write( "def f(x, y):<BR>\n" );
document.write( "    if 0 < x < 8 and 0 < y < 4:<BR>\n" );
document.write( "        return 1/32<BR>\n" );
document.write( "    return 0\r<BR>\n" );
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document.write( "# Define the probability condition for the integral<BR>\n" );
document.write( "# P(X + Y < 4)<BR>\n" );
document.write( "# Bounds for y: 0 to min(4, 4 - x) -> but 4-x <= 4 for x >= 0<BR>\n" );
document.write( "# Bounds for x: 0 to 4 (since 4-x must be > 0)\r<BR>\n" );
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document.write( "prob, error = integrate.dblquad(lambda y, x: 1/32, 0, 4, lambda x: 0, lambda x: 4 - x)\r<BR>\n" );
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document.write( "print(f\"Probability: {prob}\")<BR>\n" );
document.write( "print(f\"Fraction: {prob * 32}/32\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "Probability: 0.25<BR>\n" );
document.write( "Fraction: 8.0/32\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the probability $P(X + Y < 4)$ for the given joint probability density function, we can use either double integration or geometric area calculations since the distribution is uniform.\r<BR>\n" );
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document.write( "### **1. Identify the Region and PDF**<BR>\n" );
document.write( "The joint pdf is given by:<BR>\n" );
document.write( "$$f(x, y) = \frac{1}{32} \quad \text{for } 0 < x < 8, \ 0 < y < 4$$<BR>\n" );
document.write( "The total area of the sample space is $8 \times 4 = 32$, and the total volume (probability) is $32 \times \frac{1}{32} = 1$.\r<BR>\n" );
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document.write( "### **2. Define the Event Region**<BR>\n" );
document.write( "We want to find the probability of the event $X + Y < 4$, which can be rewritten as $Y < 4 - X$. <BR>\n" );
document.write( "Additionally, the variables are constrained by the original region:<BR>\n" );
document.write( "* $x > 0$<BR>\n" );
document.write( "* $y > 0$<BR>\n" );
document.write( "* $x < 8$<BR>\n" );
document.write( "* $y < 4$\r<BR>\n" );
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document.write( "Combining $y < 4 - x$ with $y > 0$, we find that $4 - x > 0$, which implies **$x < 4$**. <BR>\n" );
document.write( "Thus, the region of interest is a triangle with vertices at $(0,0)$, $(4,0)$, and $(0,4)$.\r<BR>\n" );
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document.write( "### **3. Calculation (Geometric Method)**<BR>\n" );
document.write( "Since $f(x, y)$ is a constant (uniform distribution), the probability is simply the area of the event region multiplied by the value of the pdf:<BR>\n" );
document.write( "$$\text{Area of Triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 4 \times 4 = 8$$<BR>\n" );
document.write( "$$P(X + Y < 4) = \text{Area} \times f(x, y) = 8 \times \frac{1}{32} = \frac{8}{32} = \frac{1}{4}$$\r<BR>\n" );
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document.write( "### **4. Calculation (Integration Method)**<BR>\n" );
document.write( "We can verify this by integrating the joint pdf over the triangular region:<BR>\n" );
document.write( "$$P(X + Y < 4) = \int_{0}^{4} \int_{0}^{4-x} \frac{1}{32} \, dy \, dx$$<BR>\n" );
document.write( "Evaluating the inner integral:<BR>\n" );
document.write( "$$\int_{0}^{4-x} \frac{1}{32} \, dy = \left[ \frac{y}{32} \right]_{0}^{4-x} = \frac{4-x}{32}$$<BR>\n" );
document.write( "Evaluating the outer integral:<BR>\n" );
document.write( "$$\int_{0}^{4} \frac{4-x}{32} \, dx = \frac{1}{32} \left[ 4x - \frac{x^2}{2} \right]_{0}^{4} = \frac{1}{32} \left( 16 - \frac{16}{2} \right) = \frac{8}{32} = \frac{1}{4}$$\r<BR>\n" );
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document.write( "**Final Result:**<BR>\n" );
document.write( "$$P(X + Y < 4) = 0.25$$ </SPAN>\n" );
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