document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169914>Question 1169914</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose the random variable X has a binomial(n,U)distribution where U is uniformly distributed random variable on [0,1].\r<BR>\n" );
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document.write( "Show that X is a discrete uniformly distributed random variable.<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 #851272 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.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=851272\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step:\r<BR>\n" );
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document.write( "**1. Understand the Distributions:**\r<BR>\n" );
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document.write( "* **Binomial Distribution:** X | U = u ~ Binomial(n, u). This means that given a specific value of U = u, X follows a binomial distribution with n trials and probability of success u.<BR>\n" );
document.write( "* **Uniform Distribution:** U ~ Uniform(0, 1). This means that U is a continuous random variable with a constant probability density function over the interval [0, 1].\r<BR>\n" );
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document.write( "**2. Find the Probability Mass Function (PMF) of X:**\r<BR>\n" );
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document.write( "We need to find P(X = k) for k = 0, 1, 2, ..., n.\r<BR>\n" );
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document.write( "P(X = k) = ∫ P(X = k | U = u) * f_U(u) du\r<BR>\n" );
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document.write( "where:<BR>\n" );
document.write( "* P(X = k | U = u) = (n choose k) * u^k * (1 - u)^(n - k) (binomial PMF)<BR>\n" );
document.write( "* f_U(u) = 1 (uniform PDF on [0, 1])\r<BR>\n" );
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document.write( "So,\r<BR>\n" );
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document.write( "P(X = k) = ∫[0, 1] (n choose k) * u^k * (1 - u)^(n - k) * 1 du<BR>\n" );
document.write( "P(X = k) = (n choose k) * ∫[0, 1] u^k * (1 - u)^(n - k) du\r<BR>\n" );
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document.write( "**3. Recognize the Beta Function:**\r<BR>\n" );
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document.write( "The integral ∫[0, 1] u^k * (1 - u)^(n - k) du is related to the Beta function.\r<BR>\n" );
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document.write( "The Beta function is defined as:\r<BR>\n" );
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document.write( "B(x, y) = ∫[0, 1] t^(x - 1) * (1 - t)^(y - 1) dt\r<BR>\n" );
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document.write( "In our case, x = k + 1 and y = (n - k) + 1 = n - k + 1.\r<BR>\n" );
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document.write( "So,\r<BR>\n" );
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document.write( "∫[0, 1] u^k * (1 - u)^(n - k) du = B(k + 1, n - k + 1)\r<BR>\n" );
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document.write( "**4. Relate the Beta Function to Factorials:**\r<BR>\n" );
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document.write( "The Beta function has a relation to factorials:\r<BR>\n" );
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document.write( "B(x, y) = Γ(x) * Γ(y) / Γ(x + y)\r<BR>\n" );
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document.write( "where Γ(z) is the Gamma function. For integer values, Γ(z) = (z - 1)!.\r<BR>\n" );
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document.write( "Therefore,\r<BR>\n" );
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document.write( "B(k + 1, n - k + 1) = k! * (n - k)! / (n + 1)!\r<BR>\n" );
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document.write( "**5. Substitute Back into P(X = k):**\r<BR>\n" );
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document.write( "P(X = k) = (n choose k) * [k! * (n - k)! / (n + 1)!]<BR>\n" );
document.write( "P(X = k) = [n! / (k! * (n - k)!)] * [k! * (n - k)! / (n + 1)!]<BR>\n" );
document.write( "P(X = k) = n! / (n + 1)!<BR>\n" );
document.write( "P(X = k) = 1 / (n + 1)\r<BR>\n" );
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document.write( "**6. Conclusion:**\r<BR>\n" );
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document.write( "Since P(X = k) = 1 / (n + 1) for k = 0, 1, 2, ..., n, X is a discrete uniformly distributed random variable over the set {0, 1, 2, ..., n}.\r<BR>\n" );
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document.write( "**Therefore, X is a discrete uniformly distributed random variable.**<BR>\n" );
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