document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163784>Question 1163784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a random variable with exponential distribution -1. Compute the probability for P (X>3). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854370 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=854370\">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 math\r<BR>\n" );
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document.write( "# Assuming lambda = 1<BR>\n" );
document.write( "lambda_val = 1<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "prob = math.exp(-lambda_val * x)<BR>\n" );
document.write( "print(f\"{prob=}\")\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "prob=0.049787068367863944\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To compute the probability $P(X > 3)$ for a random variable $X$ with an exponential distribution, we first identify the parameter.\r<BR>\n" );
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document.write( "### 1. Identify the Parameter ($\lambda$)<BR>\n" );
document.write( "In an exponential distribution, the parameter $\lambda$ (the rate) must be positive. The \"-1\" in your query likely refers to a rate of $\lambda = 1$ (where the dash is a separator). \r<BR>\n" );
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document.write( "The probability density function (PDF) is:<BR>\n" );
document.write( "$$f(x; \lambda) = \lambda e^{-\lambda x} \text{ for } x \ge 0$$\r<BR>\n" );
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document.write( "### 2. Formula for $P(X > x)$<BR>\n" );
document.write( "The probability that $X$ is greater than a specific value $x$ is given by the survival function:<BR>\n" );
document.write( "$$P(X > x) = \int_{x}^{\infty} \lambda e^{-\lambda t} dt = e^{-\lambda x}$$\r<BR>\n" );
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document.write( "### 3. Calculation<BR>\n" );
document.write( "Substituting $\lambda = 1$ and $x = 3$:<BR>\n" );
document.write( "$$P(X > 3) = e^{-(1)(3)}$$<BR>\n" );
document.write( "$$P(X > 3) = e^{-3}$$\r<BR>\n" );
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document.write( "Using the value of $e \approx 2.71828$:<BR>\n" );
document.write( "$$P(X > 3) \approx 0.049787$$\r<BR>\n" );
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document.write( "**Final Answer:**<BR>\n" );
document.write( "The probability is **$e^{-3}$**, which is approximately **$0.0498$** (or **4.98%**). </SPAN>\n" );
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