Let's define the random variable X to be the number of goals scored by a randomly selected team in a randomly selected game. The table below gives the probability distribution of X: 4 5 7 Probability: 0.061 0.154 0.228 0.229 0.173 0.094 0.041 0.015 0.004 0.001 Goals: 0 1 2 3 a. Compute the expected value of the random variable X. E(X)= type your answer..... b. Compute the standard deviation of the random variable X. SD (X)= type your answer..... goals goals 6 8 9
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- The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. y 1 p(x, y) 0 1 0.030 0.020 50 X 3 0.090 0.060 4 0.060 0.040 2 0 0.015 0.010 0.025 0.050 2 0.075 0.050 0.125 0.150 0.100 5 0.030 0.020 0.050 (a) What is the probability that there is exactly one car and exactly one bus during a cycle? 0.020 (b) What is the probability that there is at most one car and at most one bus during a cycle? 0.075 (c) What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = 0.220 x P(exactly one bus) = 0.200 (d) Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflow during a cycle? 0.985 X (e) Are X and Y independent rv's? Explain. O Yes, because p(x, y) = P(x) Py(Y). O Yes, because p(x, y) = Px(x) • P₂(Y). # ● No, because p(x, y) = Px(x) •…The probability distribution for the random variable, x, is x:2, 2.5,3.0,3.5,4.0 ,4.5 and P(x):0.07, 0.36, 0.21, 0.19, 0.10, 0.07 Calculate P(x >= 3.0) a. 0.57 O b. 0.19 O C. 0.83 d. 0.47 OThe annual premium for a $15,000 insurance policy against the theft of a painting is $250. If the (empirical) probability that the painting will be stolen during the year is 0.02, what is your expected return from the insurance company if you take out this insurance? Let X be the random variable for the amount of money received from the insurance company in the given year. E(X)= dollars
- The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. P(x, y) 1 2 0.015 0.010 0.025 1 0.030 0.020 0.050 2 0.075 0.050 0.125 3 0.090 0.060 0.150 0.060 0.040 0.100 0.030 0.020 0.050 What is the probability that there is exactly one car and exactly one bus during a cycle? What is the probability that there is at most one car and at most one bus during a cycle? What is the probability that there is exactly one car during a cycle? Exactly one bus? P(exactly one car) = P(exactly one bus) = - Suppose the left-turn lane is to have a capacity of five cars and one bus is equivalent to three cars. What is the probability of an overflov during a cycle?The probability distribution of the number of students absent on Mondays, is as follows: X O 1 2 3 4567 f(x) 0.02 0.03 0.26 0.34 0.22 0.08 0.04 0.01 (a) What is the probability that more than 3 students are absent. (b) Compute the expected value of the random variable X. Interpret this expected value. (c) Compute the variance and standard deviation of the random variable X. (d) Compute the expected value and variance of Y : (e) Compute the covariance Cov(X, Y) = 7X+ 3. (f) What is the value of the ratio Cov(X,Y)/Var(X) ? (g) What is the value of Cov(X, Y)/Var(X) for Y = ß0 + ß₁X ? (For this exrcise, you can use excel. No need to show all calculations, just specify the formulas and present the final values.) (a) (b) (c) (d) (e) (f) (g)The annual premium for a $20,000 insurance policy against the theft of a painting is $350. If the (empirical) probability that the painting will be stolen during the year is 0.02, what is your expected return from the insurance company if you take out this insurance? Let X be the random variable for the amount of money received from the insurance company in the given year. E(X)= dollars
- The joint probability distribution for the Random variable X and Y is shown below where X = the number of Strawberry ice-cream cones in the freezer Y = the number of Chocolate ice-cream cones in the freezer y p(x, y) 1 2 .025 .015 .010 .050 .030 .020 .075 090 .125 .050 150 060 012The probability distribution of random variable X is defined as follows: X 0 1 2 3 4 Probability 0 0.3 0.1 0.3 0.3 Find the expected value and the standard deviation of X.Senior high school students are randomly selected to gather data on the hours spent per day playing mobile games. The table below shows the probability distribution of X=hours spent playing mobile games. Probability Distribution of X Length (in hours) 0 1 2 3 4 5 6 Probability 0.012 0.062 0.281 0.326 ? 0.112 PALERMORY 0.083 A. Find the missing value for P(X=4) in the given probability distribution. ✓ [Select ] 0.241 0.214 0.142 0.124
- Complete the values of P (X = x) to give a legitimate probability distribution for the discrete random variable, whose possible values are -1, 2, 4, 5, and 6.High school students in Michigan sometimes get "snow days" in the winter when the roads are so bad that school is canceled for the day. Define the random variableX = the number of snow days at a certain high school in Michigan for a randomly selected school year. Suppose the table below gives the probability distribution of X. 1. Value x, 1. 3 4 5 6 7 8 9 10 Probability p, ? 0.19 0.14 0.10 0.07 0.05 0.04 0.04 0.02 0.01 0.01 (a) Write the event “the school year has O snow days" in terms of X. Then find its probability. (b) At this high school, if more than 5 snow days are used over the course of the year, students are required to make up the time at the end of the school year. What's the probability that a randomly selected school year will require make-up days?The probability distribution of random variable, X, is defined as follows: X 0 1 2 3 4 Probability 0 .3 .1 .3 .3 The P(X = 0) =?