b) Suppose that X is a random variable with the probability density function given by (2(1-x),0 ≤ x ≤ 1 otherwise f(x) = {20₁ Find the density function of W = 2X-1 using method of cumulative distribution function.
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- suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with density curve f(x) = (1/3)e(-x/3) for x > 0 and f(x) = 0 otherwise. What is the probability that response time exceeds 5 seconds? What is the probability that response time exceeds 10 seconds?Suppose that the probability density function of x is fx=3x2, 0<x<1 0, elsewhere Determine p(x < (1/3)), p((1/3) ≤ x < (2/3)), and p(x ≥ (2/3)) Determine the cumulative distribution function of x.
- If X is a continuous variable in the range 3 > X > 0 and its distribution function is as follows: F ( x ) = k : ( x3 + x2) find the probability density function?A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0Suppose that ƒ is a uniform joint probability density function on0 ≤ x 6 2, 0 ≤ y < 3. What is the formula for ƒ? What is theprobability that X < Y?
- Find the moment-generating function of the contin-uous random variable X whose probability density is given by f(x) =1 for 0 < x < 10 elsewhere and use it to find μ1,μ2, and σ2.If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.If the random variable T is the time to failure of a commercial product and the values of its probability den-sity and distribution function at time t are f(t) and F(t), then its failure rate at time t is given by f(t)1 − F(t). Thus, thefailure rate at time t is the probability density of failure attime t given that failure does not occur prior to time t.(a) Show that if T has an exponential distribution, thefailure rate is constant. (b) Show that if T has a Weibull distribution (see Exer-cise 23), the failure rate is given by αβt β−1.
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