1. Let f(x) = k(3x - x2) if 0 ≤ x ≤ 3 and f(x) = 0 if x < 0 or x > 3.
a. For what value of k is f a probability density function?
b. For that value of k, find P(X > 1). (c), Find the mean.
2. a. A type of light bulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of failure of these bulbs by an exponential density function with mean µ = 1000. Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours.
b. What is the median lifetime of these light bulbs? Problem description: #2, sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases. x = t3 + t, y = t2 + 2, -2 ≤ t ≤ 2.
3. a. Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.
b. Eliminate the parameter to find a Cartesian equation of the curve. x = 3t + 2, y = 2t + 3.
4. a. Eliminate the parameter to find a Cartesian equation of the curve.
b. Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. x = 12 cos θ, y = 2 sin θ, 0 ≤ θ ≤ π.