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  3. 1 identify whether the following random variables are discrete or...

Question: 1 identify whether the following random variables are discrete or...

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1) Identify whether the following random variables are discrete or continuous Let X- the number of bacteria per mm2 in a cell culture dish Let Y- the amount of glucose consumed by bacteria in a cell culture dish Let M the inner diameter of an arterial catheter Let N- the number of arterial catheters outside a specified diameter range Let P-the number of customer complaints received during a one month period Let Q-the amount of time it takes to receive 10 customer complaints Suppose you define a random variable X as the number of heads minus the number of tails in 2) three coin tosses. a. Is X continuous or discrete? b. Determine all possible values of X. c. Based on your answer from part B, provide the probability distribution function f(x)- Prob(X-x) for this random variable d. What is Prob(X> -2)? e. What is Prob(X-5)? f. Determine whether f(x) from part C is a valid probability distribution function. 3) Suppose the random variable X is the shelf life of medicine in days is a random variable with a probability density function f(x)-20,000/(x + 100)3 forx > 0; f(x)-0 otherwise a. Is X continuous or discrete? b. Find the probability that the medicine will have a shelf life of at least 200 days c. Find the probability that the medicine will have a shelf life from 80 to 120 days. d. Verify that f(x) is a valid probability density function. 4) X is a continuous random variable with probability density function f(x) 2(1 +x)/27 where 2 <x <5, and f(x)-0 for all other values of x a. Find Prob(X < 4). b. Find Prob(X> 3). c. Verify that f(x) is a valid probability density function.5) A continuous random variable X has the density function f(x) shown below. This density function f(x) consists of a rectangle of height r from 0<x<2 r and a semicircle of radius r from 2 r<x<4r f(x) a. Determine the value of r that will make this a valid probability density function. b. What is Prob(X> 2 r)? c. What is Prob(X< 2 r)? d. What is Prob(r < X<3r)? HINT: no integration is required. Instead, you can use your knowledge regarding the area of rectangles and semicircles to solve this problem. 6) A discrete random variable X has the following probability distribution function: x | Prob(X-X) 12 -2 1/8 1/16 1/4 5/16 1/16 3/16 15 a. Show that this is a valid probability distribution function. b. What is the Prob(X>-1.5)? c. What is the Prob(X > 6)?

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