Suppose X is the waiting time for a failure of a component and X>0. The probability density function of X is f(x) and the cumulative distribution function of X is F(x). The hazard rate, or hazard function, of X is defined as h(x) = f(x)/[1-F(x)], where h(x)dx represents the conditional probability of X being in the interval (x, x+dx) given that the component has not failed up to time x. Suppose X has a Weibull probability density function where β-1 Bx x f(x) x2 0; B,0>0
= 0 elsewhere
a) Find the hazard rate of X
b) Suppose it is known that B 2.25. Further suppose a random sample of n values of X is available. Find the maximum likelihood estimator of the hazard rate, for a given value of x.

(a) common ratio in the given sequence is, 3

(b) push ups will do kendall do on the 20th week

step-by-step explanation:

geometric sequence is a sequence of numbers that follows a pattern were the next term is found by multiplying by a constant called the common ratio(r)

(a)

given the sequence: 6 , 18 , 54 , 162 ,

this sequence is geometric sequence where each term is 3 times the previous term.

therefore,

common ratio(r) = 3.

(b)

a formula for nth term of geometric sequence is given by;

,   where r is the common ratio between the successive terms and is the first term,

to find how many push ups will kendall do on the 20th week.

here. n = 20th term.

in the given sequence first term

then;

simplify:

therefore, push ups will do kendall do on the 20th week