6.1: Discrete Random Variables and Expected Value 1. Determine whether the table represents a valid probability distribution and explain in each 2. The number of customers in line at a supermarket express checkout counter is a random variable with the following probability distribution 0.35 3. Explain why this probability distribution is "discrete". Be g . . .9 specrc. 1", E Number of Customers in Line b. Find the probability that more than 3 customers are waiting in line = c. Find P(X>2) = d. Compute the mean of the distribution: (xx: (1 decimal place) a. Compute the standard deviation of the distribution: 0,: (1 decimal place) f. Interpret the mean and standard deviation In the context of the situation 3. Lori is a commercial building contractor who will commit her company to one of three projects depending on her analysis of potential profits as shown here. Project A Project B Project C Profit, X P(X) Profit, X P (X ) Profit, X P(X ) 60000 0.1 0 0.2 40000 0.65 180000 0.6 210000 0.35 340000 0.35 250000 0.3 290000 0.45 a. Find the expected profit (gain/loss) for each of the three project Project A Project B Project C b. Which is the optimal choice based on the expected value calculations? Project A Project B Project C c. Disregard the expected values and consider the three projects' distributions. Which should Lori invest in if she is an optimist (i.e. always believes the best will happen)? Project A Project B Project C d. Disregard the expected values and consider the three projects' distributions. Which should Lori invest in if she is a pessimist (i.e. always believes the worst will happen)? Project A Project B Project C 4. The following problem presumes an American Roulette wheel. The casino payout for a bet on the "basket" is 6:1 (meaning for a $1 bet, you will receive back your $1 plus $6 from the casino). The following is a probability distribution for your profit for one spin of the wheel if you bet $1 on "basket". The probability of a "basket" is 5/38. Outcome Profit Probability a. Fill out the table at left. b. Calculate your expected profit for a single $1 bet on the "basket." Give 4 decimal places. c. If you decided to bet a $50 chip on the "basket," what is your expected profit? d. Explain why betting on the "basket" in American Roulette is NOT a "fair game."MAT 133: 6.1 WS 3 p.3 5. The following probability distribution is for the random variable X = the number of medications per patient at a doctor's office. Number of Probability a. Explain what must be true in order for this to be a Medications, X valid probability model. 0 0.19 1 0.39 2 0.14 3 0.06 4 0.07 UT 0.09 b. Explain why this probability distribution is considered 6 0.06 to be discrete. Be specific and use the context given. c. Compute the mean of the distribution: Hx= (3 decimal places) d. Compute the standard deviation of the distribution: ox= (3 decimal places) e. Interpret the mean and standard deviation in the context of the situation. In parts f) through h), ONE patient is randomly chosen. f. What is the probability that the patient takes more than 3 medications? g. To calculate part f), we must presume that values for "number of medications" are: (Circle one) Disjoint Not Disjoint Independent Dependent h. What is the probability that the patient does NOT take "2 medications"