. /ProcSet [ /PDF ] The PDF p(x) is the derivative of the random variable's CDF, /Matrix [1 0 0 1 0 0] Since $X\sim\mathcal{U}(0,2)$, $$f_X(x) = \frac{1}{2}\mathbb{I}_{(0,2)}(x)$$so in your convolution formula \frac{1}{4}z - \frac{5}{4}, &z \in (5,6)\\ /PieceInfo << Now let \(S_n = X_1 + X_2 + . 107 0 obj We explain: first, how to work out the cumulative distribution function of the sum; then, how to compute its probability mass function (if the summands are discrete) or its probability density function (if the summands are continuous). \,\,\,\left( 2F_Y\left( \frac{z (m-i-1)}{m}\right) +F_Y\left( \frac{z (m-i)}{m}\right) -F_Y\left( \frac{z (m-i-1)}{m}\right) \right) \right\} \\&=\sum _{i=0}^{m-1}\left( F_X\left( \frac{(i+1) z}{m}\right) -F_X\left( \frac{i z}{m}\right) \right) \left( F_Y\left( \frac{z (m-i-1)}{m}\right) +F_Y\left( \frac{z (m-i)}{m}\right) \right) \\&=2F_{Z_m}(z). 17 0 obj /XObject << /Fm1 12 0 R /Fm2 14 0 R /Fm3 16 0 R /Fm4 18 0 R >> Hence, using the decomposition given in Eq. Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. endobj What is the distribution of $V=XY$? /Resources 17 0 R of standard normal random variable. \quad\text{and}\quad Uniform Random Variable - an overview | ScienceDirect Topics By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. On approximation and estimation of distribution function of sum of independent random variables. Stat Probab Lett 34(1):4351, Modarres M, Kaminskiy M, Krivtsov V (1999) Reliability engineering and risk analysis. Building on two centuries' experience, Taylor & Francis has grown rapidlyover the last two decades to become a leading international academic publisher.The Group publishes over 800 journals and over 1,800 new books each year, coveringa wide variety of subject areas and incorporating the journal imprints of Routledge,Carfax, Spon Press, Psychology Press, Martin Dunitz, and Taylor & Francis.Taylor & Francis is fully committed to the publication and dissemination of scholarly information of the highest quality, and today this remains the primary goal. 14 0 obj \end{aligned}$$, \(A_i\cap A_j=B_i\cap B_j=\emptyset ,\,i\ne j=0,1m-1\), \(A_i\cap B_j=\emptyset ,\,i,j=0,1,..m-1,\), \(\{\cup _{i=0}^{m-1}A_i,\,\cup _{i=0}^{m-1}B_i,\,\left( \cup _{i=0}^{m-1}(A_i\cup B_i) \right) ^c\}\), $$\begin{aligned}{} & {} C_1=\text {Number of elements in }\cup _{i=0}^{m-1}B_i,\\{} & {} C_2=\text {Number of elements in } \cup _{i=0}^{m-1}A_i \end{aligned}$$, $$\begin{aligned} C_3=\text {Number of elements in } \left( \cup _{i=0}^{m-1}(A_i\cup B_i) \right) ^c=n_1n_2-C_1-C_2.