Probability Distributions A. El Dorado Community College considers a student to be full-time if he or she is taking between 12 and 18 units. The number of units X that a randomly selected El Dorado Community College full-time student is taking in the fall semester has the following distribution. # of units, X 12 13 14 15 16 17 18 P(X) 0.25 0.10 0.05 0.30 0.10 0.05 0.15 1. Show that the probability distribution for X is legitimate. 2. Make a histogram of the probability distribution. Show a sketch. 3. Calculate the expected (mean) number of units taken and interpret this value in context. 4. Compute and interpret the standard deviation of X. B. At El Dorado Community College, the tuition for full-time students is $50 per unit. That is, if T = tuition charge for a randomly selected full-time student, T = 50X. 1. Using the above probability distribution, construct one for T. Tuition charge, T $600 P(T) 0.25 2. Make a histogram of the probability distribution. How does it compare to the histogram of X. 3. Calculate the expected (mean) tuition, how does this compare to the expected number of units taken? 4. Compute and interpret the standard deviation of T, how does this compare to the standard deviation of X. 5. What conclusions can you make about the mean and standard deviation of a probability distribution when you multiply by a constant.
C. In addition to tuition charges, each full-time student at El Dorado Community College is assessed, student fees of $100 per semester. If C = overall cost for a randomly selected full-time student, C = 100 + T. 1. Fill in the following probability distribution for C Cost, C $700 P(C) 0.25 2. Make a histogram of the probability distribution. How does it compare to the histogram of X and T. 3. Calculate the expected (mean) cost, how does this compare to the expected tuition collected from B #3 above? 4. Compute and interpret the standard deviation of C, how does this compare to the standard deviation of T, from B #4 above. 5. What conclusions can you make about the mean and standard deviation of a probability distribution when you add a constant. Check Your Understanding. A large auto dealership keeps track of sales made during each hour of the day. Let X = the number of cars sold during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution has mean m x =1.1 and standard deviation s x = 0.943 1. Suppose the dealership s manager receives a $500 bonus from the company for each car sold. Let Y = the bonus received from car sales during the first hour on a randomly selected Friday. Find the mean and standard deviation of Y. 2. To encourage customers to buy cars on Friday mornings, the manager spends $75 to provide coffee and doughnuts. The manager s net profit T on a randomly selected Friday is the bonus earned minus this $75. Find the mean and standard deviation of T.
D. In part A we examined the probability distribution for the random variable X = the number of units taken by a randomly selected student at El Dorado Community College. Recall the mean and standard deviation for this probability distribution. # of units, X 12 13 14 15 16 17 18 P(X) 0.25 0.10 0.05 0.30 0.10 0.05 0.15 El Dorado Community College also has a campus downtown, specializing in just a few fields of study. Full-time students at the downtown campus take only 3-unit classes. Let Y = number of units taken in the fall semester by a randomly selected full-time student at the downtown campus. Here is the probability distribution of Y. # of units, Y 12 15 18 P(Y) 0.3 0.4 0.3 1. Calculate the expected (mean) number of units and interpret this value in context. 2. Compute and interpret the standard deviation of Y.
E. If you were to randomly select one full-time student from the main campus and one full-time student from the downtown campus and add their number of units, let S = X + Y. It is reasonable to assume that X and Y are independent, because each student was selected at random. Thus, P(S = 24) = P(X = 12 and Y = 12) = P(X = 12) P(Y = 12) = (0.25)(0.3) = 0.075. That is, there is a 0.075 probability that the sum is 24 units. Let us construct a probability distribution. We will let S = X + Y and we will assume X and Y are independent random variables. We need to consider all the possible combinations of values X and Y. Fill in the remainder of the table x i p i y i p i s i = x i + y i p i 12 0.25 12 0.3 24 (.25)(.3)=.075 12 0.25 15 0.4 27 (.25)(.4) =.10 12 0.25 18 0.3 30 13 12 25 13 15 28 13 18 14 12 14 15 14 15 15 15 16 16 16 17 17 17 18 18 18 Using the above, construct a probability distribution for S by adding the probabilities for each value of S. # of units, S 24 25 26 27 28 29 30 31 32 33 34 35 36 P(S)
1. Calculate the expected (mean) number of units and how does it compare to the means of X and Y? 2. Compute the variance of S, how does it compare to the variances of X and Y? 3. Compute the standard deviation of S. Compare it to the standard deviation of X and Y, do you get the same conclusions as in previous question? 4. What conclusions can you make about the mean, standard deviation and variance of a probability distribution that is the sum of two random variables? Check Your Understanding A large auto dealership keeps track of sales and lease agreements made during each hour of the day. Let X = the number of cars sold and Y = the number of cars leased during the first hour of business on a randomly selected Friday. Based on previous records, the probability distribution of X has mean m x =1.1 and standard deviation s x = 0.943 and the probability distribution of Y has mean m y = 0.7 and standard deviation s y = 0.64 Define T = X + Y 1. Find and interpret m t. 2. Compute s t assuming X and Y are independent. 3. The dealership s manager receives a $500 bonus for each car sold and a $300 bonus for each car leased. Find the mean and standard deviation of the manager s total bonus.
Summary: Effect of a Linear Transformation on the Mean and Standard Deviation If Y = a + bx is a linear transformation of the random variable X, then The probability distribution of Y has the same shape as the probability distribution of X. µ Y = a + bµ X. σ = b σ (since b could be a negative number). Mean of the Sum or Difference of Random Variables For any two random variables X and Y, if D = X ± Y, then the expected value of D is E(D) = µ D = µ X ± µ Y In general, the mean of the sum or difference of several random variables is the sum or difference of their means. (With the Variance of the Sum or Difference of Random Variables For any two random variables X and Y, if D = X ± Y, then the variance of D is In general, the variance of the sum or difference of two
Check Your Understanding 1. Let B = the amount spent on books in the fall semester for a randomly selected fulltime student at El Dorado Community College. Suppose that B 153and B 32. Recall from earlier that C = overall cost for tuition and fees for a randomly selected fulltime student at El Dorado Community College and that C = $832.50 and C = $103. Find the mean and standard deviation of the cost of tuition, fees, and books (C + B) for a randomly selected full-time student at El Dorado Community College. 2. Let us define T = amount collected from a randomly selected full-time student at the main campus and U = amount collected from a randomly selected full-time student at the downtown campus. The means and standard deviations are t = $732.50 t = $103 u = $825 u = $126.50 Suppose we randomly select one full-time student from each of the two campuses. What are the mean and standard deviation of the difference in tuition charges, T U? Interpret each of these values.
F. What happens if we combine two Normal (continuous) random variables? * Any sum or difference of independent Normal random variables is also Normally distributed. * The mean and standard deviation of the resulting Normal distribution can be found using the same rules for means and variances that we have been using for discrete random variables. Check Your Understanding 1. Suppose that a certain variety of apples have weights that follow a Normal distribution with a mean of 9 ounces and a standard deviation of 1.5 ounces. If bags of apples are filled by randomly selecting 12 apples, what is the probability that the sum of the weights of the 12 apples is less than 100 ounces? 2. To save time and money, many single people have decided to try speed dating. At a speed dating event, women sit in a circle and men spend about 10 minutes getting to know a woman before moving on to the next one. Suppose that the height M of male speed daters follows a Normal distribution with a mean of 69.5 inches and a standard deviation of 4 inches and the height F of female speed daters follows a Normal distribution with a mean of 65 inches and a standard deviation of 3 inches. What is the probability that a randomly selected male speed dater is taller than the randomly selected female speed dater with whom he is paired?