A/ Determine the cdf of an exponential distribution with pdf shown below. Find the probability when x>2 at 2=3. x >0 f (x)={2 otherwise
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- a. What is the probability that the lifetime X of the first component exceeds 3? b. What are the marginal pdf's of X and Y? Are the two lifetimes independent? x. What is the probability that the lifetime of at least one component exceeds 3?Suppose X1, . . . , Xn ∼ Exponential(λ) is a set of n observations drawn independently from an Exponential distribution. (e) Take the second partial derivative of the score function. (f) Check to make sure this value is negative to ensure that the log-likelihood function is concave down.Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?
- Suppose that X has a lognormal distribution with parameters θ = 5 and ω2 = 9. Determine the following: a. P(X < 13,300)b. Value for x such that P(X ≤ x) = 0.95c. Mean and variance of XSuppose a continuous random variable X pdf constant from 0 to 7.516, where the pdf is zero elsewhere. what is variance of this random variable?1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.
- Find the probability that the range of a random sample of size 4 from theuniform distribution having the pdf f(x) = 1, 0 < x < 1, zero elsewhere, is lessthan 12 .Suppose that Y has an exponential distribution with mean Beta. Show that 2Y/Beta has a chi square distribution with 2 degrees of freedom.Suppose that X is a continuous unknown all of whose values are between -5 and 5 and whose PDF, denoted f, is given by f ( x ) = c ( 25 − x^2 ) , − 5 ≤ x ≤ 5 , and where c is a positive normalizing constant. What is the expected value of X^2?
- Suppose that the pdf of a random variable is given by f(y)=1/10 , -5<y<5 For what value of y does the cdf equals 0.6?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?b) Explain how X can be related to the exponential distribution by using a moment generating function argument.