Which of the following is true about the probability density function (pdf) of a continuous random variable?
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- Verify that the function corresponding to the following figure is a valid probability density function. Then find the following probabilities:What is the expected value of a continuous random variable X with probability density function (pdf) given by f(x) = 2x, 0 < x < 1?Let X be a random variable whose probability density function is plotted below. If one was to generate X by passing a uniform (0, 1) random variable through a function, then what function would you use? (your answer should be a function of x). 0.9- 0.8- 0.7- 0.6- 0.5 qula Snip 0.4 0.3- 0.2- 0.1 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
- The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution. (a) Identify the graph of the uniform density function. (b) What is the probability of generating a number between 0.82 and 0.98? (c) What is the probability of generating a number greater than 0.94? ... (a) Choose the correct graph of the uniform density function below. A. В. OC. ADensity 1.2- ADensity 1.2- 1- ADensity 1.2- 1 1. 0.8- 0.6- 0.4- 0.2- 0- 0.8- 0.6- 0.4- 0.2- 0- 0.8- 0.6- 0.4- 0.2- 0+ X 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 (b) The probability is (Simplify your answer.) (c) The probability is (Simplify your answer.)Why is P(X = c) = 0 for any number c in the domain of a probability density function?The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution. (a) Identify the graph of the uniform density function. (b) What is the probability of generating a number between 0.48 and 0.62? (c) What is the probability of generating a number greater than 0.85? (a) Choose the correct graph of the uniform density function below. O A. O B. Density 1.2- 1- 0.8- 6. 0.6- 0 0.2 0.4 0.6 0.8 11.2 0.4- 0.2- to (b) The probability is (c) The probability is Q M (Simplify your answer.) (Simplify your answer.) ADensity 1.2- 1- 0.8- 0.6- 0.4- 0.2- to 0 0.2 0.4 0.6 0.8 1 1.2 Q M O C. 1.2- 1- 0.8- 0.6- 0.4- 0.2- Density 0+ 0 0.2 0.4 0.6 0.8 1 12 Q Q
- Students arrive at a lecture theatre independently. Suppose the number of students arriving in an hour follows a Poisson distribution with mean 10. Let 7 (in hours) be the time required to wait for 5 students to arrive. Derive the probability density function of T.StatQ10The random-number generator on calculators randomly generates a number between 0 and 1. The random variable X, the number generated, follows a uniform probability distribution. (a) Identify the graph of the uniform density function. (b) What is the probability of generating a number between 0.37 and 0.69? (c) What is the probability of generating a number greater than 0.83? ... OA. В. OC. Density 1.2- A Density 1.2- ADensity 1.2- 1- 1. 1- 0.8- 0.8- 0.6- 0.4- 0.2- 0+ 0.8- 0.6- 0.6- 0.4- 0.2- 04 0 0.2 0.4 0.6 0.8 1 0.4- 0.2- X 0- 0.2 0.4 0.6 0.8 1 1.2 1.2 0.2 0.4 0.6 0.8 1 1.2 (b) The probability is (Simplify your answer.) (c) The probability is (Simplify your answer.) B.