The lifetimes, in months, of two components in a system, denoted X and Y, have joint probability density function S 4xye-(2x+y) x > 0 and y > 0 f(x) = otherwise What is the probability that both components last longer than one month? a. b. Find the marginal probability density functions fx(x) and fy(y). Are X and Y independent? Explain. C.

Section: Probabilty Density Functions

2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs 2 S (v) . else What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density а. function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: Find?(r - 1.5)) = 1 CO0

1 Given the probability density function f(x) = over the interval [3,6], find the expected value, the mean, 3 the variance and the standard deviation. Expected value: Mean: Variance: Standard Deviation: > Next Question

A density curve consists of the line segment connecting the points (0,1) and (0.5,1) and the segment connecting(0.5, 1) to the x-axis.a. Determine the coordinate point where the second segment crosses the x-axis.b. Determine the slope of that segmentc. Determine the equation of the line containing this segment (y = mx + b)d. Calculate the probability P(X > 1)

Let X be the total amount saved in dollars at the cash register. The pdf of X is given by: Value of X 20 28 40 50 58 70 probability 0.36 0.21 0.24 0.1225 0.0175 0.03 .0175 0.0025 Before I make the purchase, you would place my chance of saving $20 at 24%. Suppose you approach me after I pay for my items and ask: "Did you save any money?" If I answer "no", then you know I didn't save $20 and thus the chance of saving $20 changes from the unconditional chance 24% to the conditional chance 0%. However, if I answer "yes" my chance of saving $20 would get higher. What would be the conditional chance of saving $20 if you knew I got a discount? Mathematically, we seek: P(X = 20|X > 0 ). Even though it looks weird, it is just a conditional probability. Find it. P(X = 20|X > 0) = P(X = 20 and X > 0) / P(X > 0) = Hint: P(X = 20 and X > 0) be read off the table by identifying possible values where the event (X = 20 and X > 0) is true, don't use the multiplication rule. Do not round, express your…