Problem 3 (Needed for Problem 4) A continuous random variable X is said to have an...
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
Problem 4: Memoryless Property of Exponential Random Variable The lifetime of a stream of electrons injected in a p-type semiconductor follows an exponential distribution with a mean value of 1 ms. Assuning that an electron injected in this semiconductor has survived for 2 ms, what is the probability that this electron survives for an additional 1 ms?
2. Let X be an exponential random variable with rate A > 0. In this problem you will show that X satisfies the memoryless property. Let s 2 0 and t > 0. Show that P(X > t + s| X > s) = e-M
exponential distribution 3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution of Brown's future lifetime is Y, an exponential random variable with mean B. Smith and Brown have future lifetimes that are independent of one another. Find the probability that Smith outlives Brown. Answer #3: (D) a (E) (A) (B) (C) 3. The distribution of Smith's future lifetime is X, an exponential random variable with mean a, and the distribution...
Problem #3. X is a random variable with an exponential distribution with rate 1 = 3 Thus the pdf of X is f(x) = le-ix for 0 < x where = 3. a) Using the f(x) above and the R integrate function calculate the expected value of X. b) Using the dexp function and the R integrate command calculate the expected value of X. c) Using the pexp function find the probability that .4 SX 5.7 d) Calculate the probability...
Suppose T is a continuous random variable whose probability is determined by the ex- ponential distribution, f(t), with mean μ. a. Compute the probability that T is less than p b. The median of a continuous random variable T is defined to be the number, m, such that P(T which mIn other words, if f(t) is the PDF of T, it is the number m for P(T )f(t) dt Compute the median for the exponential random variable T above. Is...
Problem 5. Let X be a continuous random variable with a 2-paameter exponential distribution with parameters α = 0.4 and xo = 0.45, ie, ;x 2 0.45 x 〈 0.45 f(x) = (2.5e-2.5 (-0.45) Variable Y is a function of X: a) Find the first order approximation for the expected value and variance of Y b) Find the probability density function (PDF) of Y. c) Find the expected value and variance of Y from its PDF Problem 5. Let X...
4 (3 points) Suppose a random variable X has the following probability density function: 3x2 -1srs0 0 otherwise f(x) (a) Compute Pr[Xs-1/2 (b) Compute E (X), the expectation of x (c) Compute the cumulative distribution function of this random variable (for all real numbers).
5. (15 Points) Let T be a random variable that is the time to failure (in years) of certain type of electrical component. T has an exponential probability density function f(x,A) =e, if >0 10, otherwise. Compute the probability that a given component will fail in 5 years or less. 5. (15 Points) Let T be a random variable that is the time to failure (in years) of certain type of electrical component. T has an exponential probability density function...
Problem 5. Suppose that the continuous random variable X has the distribution fx(z),-oo < x < oo, which is symmetric about the value x-0. Evaluate the integral: Fx (t)dt -k where Fx(t) is the CDF for X, and k is a non-negative real number.