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- Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?Suppose x is a random variable best described by a uniform probability that ranges from 1 to 6. Compute the followingQ2) The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the below density function , Find the probability that a bottle of this medicine will have a shelf life of anywhere between 60 and 120 days 20000 0Battery life between charges for the Motorolla Droid Razr Maxx is 20 hours when the primary use is talk time. The battery life drops to 7 hours when the phone is primarily used for Internel applications over cellular. Assume that the battery life in both cases follows an exponential distribution. a. Show the probability density function for battery life for the Droid Razr Maxx phone when its primary use is talk time. b. What is the probability that the battery charge for a randomly selected Droid Razr maxx phone will last no more than 15 hours when its primary use is talk time? c. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last more than 20 hours when its primary use is talk time? d. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last no more than 5 hours when its primary use is Internet Applications?Show that the probability density function of a negative binomial random variable equals theprobability density function of a geometric random variable when r = 1. Show that the formulasfor the mean and variance of a negative binomial random variable equal the corresponding resultsfor a geometric random variable when r = 1.2. Let X be a random variable with density function ) = √(2x² - + f(x) = Find the expected value of g(x) = 2X+1. 8x2 +5x + 6), 1 < x < 2, 0, elsewhere