If Y has a geometric distribution with probability of success p, show that the moment-generating function for Y is

  M₀ (t)  = p / e^-t -q = p (e^-t -q) ^ -1

Step-by-step explanation:

Let the random variable Y have a geometric distribution g (y;p) = pq y-¹

The m.g.f of the geometric distribution is derived as below

By definition , M₀ (t) = E (e^ ty) = ∑ (e^ ty )( q ^ y-1)p    ( for ∑ , y varies 1 to infinity)

                   = pe^t ∑(e^tq)^y-1

                    = pe^t/1- qe^t,     where qe^t <1

In order to differentiate the m.g.f we write it as

           M₀ (t)  = p / e^-t -q = p (e^-t -q) ^ -1

 M₀` (t)  = pe^-t (e^-t -q) ^ -2  and

 M₀^n(t)  = 2pe^-2t (e^-t -q) ^ -3   - pe^-t (e^-t -q) ^ -2  

Hence

E (y) = p (1-q)-² = 1/p

E (y²) =2 p (1-q)-³ - p (1-q)-²

        = 2/p² - 1/p  and

σ² = [E (y²) -E (y)]²

     = 2/p² - 1/p - (1/p)²

      = q/p²

step-by-step explanation:

i found the life tables for the year 2012. 

life table of males at age 30. probability of dying between ages x and x+1

total population:  

29–30 . . . . . . . . . . . . 0.001030

30–31 . . . . . . . . . . . . 0.001061

males:  

29–30 . . . . . . . . . . . . 0.001420

30–31 . . . . . . . . . . . . 0.001448

white population

29–30 . . . . . . . . . . . . 0.001005

30–31 . . . . . . . . . . . . 0.001037

white males

29–30 . . . . . . . . . . . . 0.001372

30–31 . . . . . . . . . . . . 0.001399

non hispanic white population

29–30 . . . . . . . . . . . . 0.001083

30–31 . . . . . . . . . . . . 0.001116

non hispanic white males

29–30 . . . . . . . . . . . . 0.001477

30–31 . . . . . . . . . . . . 0.001503

jacob, non hispanic white male. 30 years old.

probability that jacob lives to be 30 years old. 

since the given probabilities are probabilities of dying, i think we simply subtract it from 1 to get the probability of living. we will use the probability of non hispanic white male.

                                          probability of living to be 30 y.o.

29 - 30 : 1 - 0.001477 = 0.998523

30 - 31 : 1 - 0.001503 = 0.998497

1/3 + 1/3 ≠ 2/6 because 1/3 alone = 2/6  

step by step explanation:

1/3 + 1/3 = 2/3       2/3=4/6