document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195038>Question 1195038</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find random variable X+2Y and Y median, dispersion and correlation coefficient, if X and Y are independent variables, X~P(4)(Poisson variable) and Y~T(1, 6)(continuous distribution) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848501 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848501\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's analyze the random variable X + 2Y and find the requested statistics.\r<BR>\n" );
document.write( "\n" );
document.write( "**1. Define the Distributions**\r<BR>\n" );
document.write( "\n" );
document.write( "* **X:** Poisson distribution with mean (λ) = 4. <BR>\n" );
document.write( "    * Probability Mass Function (PMF): P(X = k) = (e^(-λ) * λ^k) / k! <BR>\n" );
document.write( "* **Y:** Continuous uniform distribution over the interval [1, 6].<BR>\n" );
document.write( "    * Probability Density Function (PDF): f(y) = 1/5 for 1 ≤ y ≤ 6, and 0 otherwise.\r<BR>\n" );
document.write( "\n" );
document.write( "**2. X + 2Y**\r<BR>\n" );
document.write( "\n" );
document.write( "* Since X and Y are independent, the distribution of X + 2Y will not have a simple closed-form expression. <BR>\n" );
document.write( "* However, we can simulate values of X and Y and then calculate values of X + 2Y to analyze its properties.\r<BR>\n" );
document.write( "\n" );
document.write( "**3. Median, Dispersion, and Correlation Coefficient of Y**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Median of Y:** <BR>\n" );
document.write( "    * For a continuous uniform distribution, the median is the midpoint of the interval: (1 + 6) / 2 = 3.5\r<BR>\n" );
document.write( "\n" );
document.write( "* **Dispersion of Y (Standard Deviation):**<BR>\n" );
document.write( "    * For a continuous uniform distribution over the interval [a, b], the standard deviation is: <BR>\n" );
document.write( "        * σ_Y = (b - a) / √12 = (6 - 1) / √12 ≈ 1.44\r<BR>\n" );
document.write( "\n" );
document.write( "* **Correlation Coefficient between X and Y:**<BR>\n" );
document.write( "    * Since X and Y are independent, their correlation coefficient is 0.\r<BR>\n" );
document.write( "\n" );
document.write( "**Simulation in Python**\r<BR>\n" );
document.write( "\n" );
document.write( "```python<BR>\n" );
document.write( "import numpy as np<BR>\n" );
document.write( "from scipy.stats import poisson, uniform\r<BR>\n" );
document.write( "\n" );
document.write( "# Define the distributions of X and Y<BR>\n" );
document.write( "X = poisson.rvs(mu=4, size=10000)  # Poisson distribution with mean 4<BR>\n" );
document.write( "Y = uniform.rvs(loc=1, scale=5, size=10000)  # Uniform distribution between 1 and 6\r<BR>\n" );
document.write( "\n" );
document.write( "# Create the new random variable Z = X + 2Y<BR>\n" );
document.write( "Z = X + 2*Y\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate the median of Y<BR>\n" );
document.write( "median_y = np.median(Y)\r<BR>\n" );
document.write( "\n" );
document.write( "# Calculate the dispersion of Y (standard deviation)<BR>\n" );
document.write( "dispersion_y = np.std(Y)\r<BR>\n" );
document.write( "\n" );
document.write( "# Since X and Y are independent, their correlation coefficient is 0<BR>\n" );
document.write( "correlation_coefficient = 0\r<BR>\n" );
document.write( "\n" );
document.write( "# Print the results<BR>\n" );
document.write( "print(f\"Median of Y: {median_y}\")<BR>\n" );
document.write( "print(f\"Dispersion of Y (Standard Deviation): {dispersion_y}\")<BR>\n" );
document.write( "print(f\"Correlation Coefficient between X and Y: {correlation_coefficient}\")<BR>\n" );
document.write( "```\r<BR>\n" );
document.write( "\n" );
document.write( "This code will generate a large number of samples from the distributions of X and Y, and then calculate the median, dispersion, and correlation coefficient of Y. \r<BR>\n" );
document.write( "\n" );
document.write( "**Key Points:**\r<BR>\n" );
document.write( "\n" );
document.write( "* The distribution of X + 2Y will be more complex than either X or Y individually due to the combination of a discrete (Poisson) and continuous (uniform) distribution.<BR>\n" );
document.write( "* The median of Y for a uniform distribution is easily calculated as the midpoint of the interval.<BR>\n" );
document.write( "* The independence of X and Y ensures a correlation coefficient of 0.\r<BR>\n" );
document.write( "\n" );
document.write( "I hope this explanation is helpful! Let me know if you have any further questions.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );