Show that the following functions are probability density functions for some value of k and determine k. Then, determine the mean ((a),(c),(e),(g)) and variance ((b),(d),(f),(h)) of X. Round your answers to three decimal places (e. g. 98.765).

(a),(b)f (x) = kx2 for 0 < x < 23
(c),(d)f (x) = k(1 + 2x) for 0 < x < 20
(e),(f)f (x) = ke-x for 0 < x
(g),(h)f (x) = k where k > 0 and 110 < x < 110 + k

(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)

a) 17.5

b) 15.6

c) 13.3

d) 21.51

Step-by-step explanation:

The given function is equal to:

f(x)=kx^2

where

where y=23

Clearing k=0.00025

a)

b)

c) The function is equal to:

f(x)=k(1+2x)

where y=20

k=0.0024

d)

arithmetic sequence

step-by-step explanation:

given that a sequence has terms as

19, 15, 11,

this is represented on a graph

the pattern indicates that the sequence is an arithmetic sequence with i term

a=19 and common difference d =-4

any general term

thus the sequence is reduced by 4 for each successive term starting from 1 and nth term is

24-5n.

first negative term would start after 5th term

5th term =24-25< 0

from 5th term onwards the sequence would be negative.

there are many solutions.

step-by-step explanation:

step 1: simplify both sides of the equation.

4x+6=4x^2+10x−4

step 2: subtract 4x^2+10x-4 from both sides.

4x+6−(4x^2+10x−4)=4x^2+10x−4−(4x^2+10x−4)

−4x^2−6x+10=0

step 3: factor left side of equation.

−2(2x+5)(x−1)=0

step 4: set factors equal to 0.

x=-5/2 or x=1