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- 3. Let X be a scalar random sample from the following density f(x) = 2(0-x) 82 Construct a (1-a) Cl for 0. for 0 ≤ x ≤ 0The dealer's profit, in units of $1000, on a new automobile is a random variable X having the density function given by ={2(1-x), 0Find the mean, standard deviation and the median of the density function: f(x)= (6/343)x(7-x) [0,7]Suppose that X, Y are independent standard normal RVs. Find the joint pdf of Z, W where Z = XY, W = 3X – 2Y.Let the random variable X follow a uniform distribution in the interval [0,0]. Then the density function of X is if 0Consider a group of people who have received treatment for a disease such as cancer. Let t be the survival time, the number of years a person lives after receiving treatment. The density function giving the distribution of t is p(t) = Ce-Ct for some positive constant C, and the cumulative distribution function is P(t) = , p(x)dx. Think carefully about what the practical meaning of P(t) is, being sure that you can put it into words. a) The survival function, S(t), is the probability that a randomly selected person survives for at least t years. Find a formula for S(t). b) Suppose that a patient has a 80 percent chance of surviving at least 2 years. Find C. c) Using the value of C you found in (b), find the probability that the patient survives up to (that is, less than or equal to) 1 years. d) Using the value of C you found in (b), find the the mean survival time for patients with this survival function, in years.The cumulative distribution function of the continuous uniform distribution between con- stants a and b is given by F(x) = P(X ≤ x) = x-a - a x b (a) The probability density function is f(x) = F(r). Find the form of f(x). (b) Find the derivative of f(x) for x = [a, b]. Is f(x) decreasing, increasing or flat in this region? (c) Does f(x) have a single maximum in the region [a,b]? If so, what is it, or if not, why not?(i) Continuous RVs X and Y have a joint PDF (m+n+2)! S(x, y) = (1-x)"y". m!n! for 0 < yExercise 24 Let A and B be 2 events : P(A) : 1 and P(B') 1 Can A and B be mutually exclusive? Why?Let W ~ Gamma(100, 3.4). (a) Give the exact value of P(1.5 < W < 5.0) (b) Give the approximate value of P(1.5 < W < 5.0) using normal approximationConsider the probability density fy (x) = a . e~b lx!, where X is random variable whose allowable values range from x = - ∞ to + o. Find (a) the CDF (b) the relation between a and b (c) the probability that x lies between 1 and 2.Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON