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- (b) (i) Briefly explain how moment generating function of a random variable, y may be used to Generate its moments. (ii) Let the random variable, y have probability density function S(t) = kye-1/2y y20 , elsewhere Where k> 0 is a constant. Find the moment generating function of y and use it to evaluate the mean and variance of y leaving your results in terms of k.If two balanced dice are rolled simultaneously, X₂ is the appearance of one side on the first die, and X₂ represents the number of appearances on both sides of the die a Find the joint probability mass function (tmp), PCX₁ = X₁₁ X₂ = X/₂₁ ) ! b Find the marginal probability mass function for each of the variables X₁ and X₂!4. The monthly revenue (in millions) of a local gaming company is represented by a continuous random variable X having the probability density f(x) = (1 - (x-1)²), 0 < x < 2, 0, elsewhere Find the mean and variance of X.
- 3. The probability density functions of two statistically independent random variables X and Y are fx(x) = }u(x - 1)e-lx-1)/2 fr(y) =u(y- 3)e-(s-3)/4 %3D Find the probability density of the sum W =X+Y.If X and Y are independent random variables with common density f (x) = for 0 < x < 2 then (fx * fy) (3)Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
- The bearing capacity of the soil under a foundation is measured to range from 200 kPA to 450 kPA. The probability density within this range is given by engineering office as f(x) = 3 ,200< x < 450 450 ,otherwise If you know that the column on which stands is designed to carry a load 275 kPA, what is the probability of the failure of the foundation? Do you think redesigning project is necessary?The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…Suppose that the amount of time a hospital patient must wait for a nurse's help is described by a continuous random variable with density function f(t) = e-t/3 where t≥ 0 is measured in minutes. (a) What is the probability that a patient must wait for more than 4 minutes? (b) A patient spends a week in the hospital and requests nurse assistance once each day. What is the probability that the nurse will take longer than 5 minutes to respond on (exactly) two occasions? (c) What is the probability that on at least one call out of seven, the nurse will take longer than 7 minutes to respond?
- Assume the time required to fully recharge an electric car is uniformly distributed between 160 and 220 minutes. What is the expected charge time, E(x)?Let X and Y be two random variables with joint probability mass function: p(x,y) = xy(1+y) for X=1,2,3 and Y=1,2 p(x,y) = 0, Otherwise. Please enter the answer to 2 decimal places. • What is the variance of (4-5X)?
