Question

6.3 cast a die a number of independent times until a six appears on the up side of the die. (a) find the pmf ???????????????? of ????, the number of casts needed to obtain that first six. (b) show that ∑???????????????? ???????????????? ???? 1. (c) determine ???????????? ???? 1, 3, 5, 7, ... ????. (d) find the cdf ???????????????? ???? ???????????? ???? ????????

Answer 1

hello your question is incomplete attached below is the complete question

Answer : A)
`P_(x) (x) = P ( X = x )\\= ((1)/(6))((5)/(6))^(x-1)`

B) attached below

C) P ( x = 1,3,5,7,----) = 6/11

D) F(x) = 1 - (5/6)^x

Explanation:

A) probability of 6 appearing = 1/6

probability of 6 not appearing = 5/6

number of times a die is thrown = X

Hence PMF ( probability of mass function )

`P_(x) (x) = P ( X = x )\\= ((1)/(6))((5)/(6))^(x-1)`

where number of failures = x-1 , number of trials ( number of times a die is thrown ) = x

B) show that attached below

C) determine : P(X= 1,3,5,7..) ( attached below )

therefore P ( x = 1,3,5,7,----) = 6/11

d) find the CDF of F(x) = P( X ≤ x )

F(x) = 1 - (5/6)^x