15. A random variable X has probability density function 2x, 0 0.75 / X > 0.5).

6.The continuous random variable X has the probability density function given by 1 -2 < x < 4 f(x) = otherwise Determine P(X < 2)

13. Suppose a continuous random variable x has the probability density Sk(1+ x), 0 1.4) using distribution function, determine the probabilities that (d) x is less than 2.2 (e) between 1.4 and 1.6 (f) Calculate mean and variance for the probability density function.

I. The length of time required by students to complete a 1-hour exam is a random variable X (in hours) with probability density function f(x) = 45(x3 + 1) for 0 < x < 1 and f(x) = 0 for x • • • • • • otherwise. Randomly select a student from this class. Whatistheprobabilitythatthecompletiontimeforthisstudentisatleast50minutes? Note that 40 minutes is equal to 40/60 hr. What is the probability that completion time is between 20 and 40 minutes? Determine the mean and variance of completion time. What is the CDF Graph the CDF Graph the PDF II . The average American eats 2,000 calories per day with a standard deviation of 500 calories. If this distribution follows a normal distribution, (a) What is the probability that someone eats between 2,000 and 3,000 calories per day? (b) How many calories corresponds to the 70th percentile? III. Draw a picture (Venn Diagram) for the following distributions. Additionally, find the fol- lowing probabilitis if P(A) = 0.4 and P(B) = 0.3 and P(A ∩…