document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178326>Question 1178326</A>: <I><SPAN STYLE=\"word-break: break-all;\"> After watching a number of children playing games at a video arcade, a statistician estimated the following probability distribution of X, the number of games per visit.<BR>\n" );
document.write( "X 1 2 3 4 5 6 7<BR>\n" );
document.write( "P(x) .05 .15 .15 .25 .20 .10 .10<BR>\n" );
document.write( "a. Determine the legitimacy of the distribution of the number of games per visits.<BR>\n" );
document.write( "b. What is the probability that a child will play more than four games per visits?<BR>\n" );
document.write( "c. What is the probability that a child will play at least two games per visits?<BR>\n" );
document.write( "d. What is the expected number of games per visits?<BR>\n" );
document.write( "e. What is the variance and standard deviation of the distribution?<BR>\n" );
document.write( "f. Compute the coefficient of variation for the distribution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850353 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=850353\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze this probability distribution step-by-step.\r<BR>\n" );
document.write( "\n" );
document.write( "**a. Legitimacy of the Distribution**\r<BR>\n" );
document.write( "\n" );
document.write( "For a probability distribution to be legitimate, two conditions must be met:\r<BR>\n" );
document.write( "\n" );
document.write( "1.  **All probabilities must be between 0 and 1 (inclusive).**<BR>\n" );
document.write( "2.  **The sum of all probabilities must equal 1.**\r<BR>\n" );
document.write( "\n" );
document.write( "Let's check:\r<BR>\n" );
document.write( "\n" );
document.write( "* All probabilities are between 0 and 1.<BR>\n" );
document.write( "* Sum of probabilities: 0.05 + 0.15 + 0.15 + 0.25 + 0.20 + 0.10 + 0.10 = 1.00\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the distribution is legitimate.\r<BR>\n" );
document.write( "\n" );
document.write( "**b. Probability of More Than Four Games**\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find P(X > 4), which is P(X = 5) + P(X = 6) + P(X = 7).\r<BR>\n" );
document.write( "\n" );
document.write( "* P(X > 4) = 0.20 + 0.10 + 0.10 = 0.40\r<BR>\n" );
document.write( "\n" );
document.write( "**c. Probability of At Least Two Games**\r<BR>\n" );
document.write( "\n" );
document.write( "We want to find P(X ≥ 2), which is 1 - P(X < 2) = 1 - P(X = 1).\r<BR>\n" );
document.write( "\n" );
document.write( "* P(X ≥ 2) = 1 - 0.05 = 0.95\r<BR>\n" );
document.write( "\n" );
document.write( "**d. Expected Number of Games (Mean)**\r<BR>\n" );
document.write( "\n" );
document.write( "The expected value (E[X]) is the mean of the distribution:\r<BR>\n" );
document.write( "\n" );
document.write( "* E[X] = Σ[x * P(x)]<BR>\n" );
document.write( "* E[X] = (1 * 0.05) + (2 * 0.15) + (3 * 0.15) + (4 * 0.25) + (5 * 0.20) + (6 * 0.10) + (7 * 0.10)<BR>\n" );
document.write( "* E[X] = 0.05 + 0.30 + 0.45 + 1.00 + 1.00 + 0.60 + 0.70 = 4.10\r<BR>\n" );
document.write( "\n" );
document.write( "**e. Variance and Standard Deviation**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Variance (Var[X]):**<BR>\n" );
document.write( "    * Var[X] = E[X²] - (E[X])²<BR>\n" );
document.write( "    * E[X²] = Σ[x² * P(x)]<BR>\n" );
document.write( "    * E[X²] = (1² * 0.05) + (2² * 0.15) + (3² * 0.15) + (4² * 0.25) + (5² * 0.20) + (6² * 0.10) + (7² * 0.10)<BR>\n" );
document.write( "    * E[X²] = 0.05 + 0.60 + 1.35 + 4.00 + 5.00 + 3.60 + 4.90 = 19.50<BR>\n" );
document.write( "    * Var[X] = 19.50 - (4.10)² = 19.50 - 16.81 = 2.69\r<BR>\n" );
document.write( "\n" );
document.write( "* **Standard Deviation (σ):**<BR>\n" );
document.write( "    * σ = √Var[X] = √2.69 ≈ 1.6401\r<BR>\n" );
document.write( "\n" );
document.write( "**f. Coefficient of Variation (CV)**\r<BR>\n" );
document.write( "\n" );
document.write( "The coefficient of variation (CV) is the ratio of the standard deviation to the mean, expressed as a percentage:\r<BR>\n" );
document.write( "\n" );
document.write( "* CV = (σ / E[X]) * 100<BR>\n" );
document.write( "* CV = (1.6401 / 4.10) * 100 ≈ 40.00%\r<BR>\n" );
document.write( "\n" );
document.write( "**Results**\r<BR>\n" );
document.write( "\n" );
document.write( "* **a. Legitimacy:** The distribution is legitimate.<BR>\n" );
document.write( "* **b. P(X > 4):** 0.40<BR>\n" );
document.write( "* **c. P(X ≥ 2):** 0.95<BR>\n" );
document.write( "* **d. E[X]:** 4.10 games<BR>\n" );
document.write( "* **e. Var[X]:** 2.69, σ: 1.6401<BR>\n" );
document.write( "* **f. CV:** 40.00%<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );