Let X be the random variable the number of fours observed when two dice are rolled

together once. Show that X is a discrete random variable.

4. The pmf of a discrete random variable X is given by

P(X ? x) ? kx for x ?1,2,3, 4,5,6

Find the value of the constant k,

P?X ? 4?

and

P?3 ? X ? 6?

5. A fair coin is flip until a head appears. Let N represent the number of tosses required to

realize a head. Find the pmf of N

6. A discrete random variable Y has a pmf given by

( ) ? ? for 0,1,2,.....

4

3 P Y ? y ? c y ?

x

Find the value of the constant c and

P?X ? 3?

7. Verify that

x k

k k

x

for 0,1,2,.....

( 1)

2

f(x) ?

?

?

can serve as a pmf of a random variable X.

8. For each of the following determine c so that the function can serve as a pmf of a random

variable X.

a)

f(x) ? cx for x ?1,2,3,4,5

b)

f(x) cx for x 0,1,2,.....k

2

? ?

c)

f(x) ? ? for 0,1,2,3.....

6

1 ? c x ?

x

d)

f(x) ? 2 for ? for ? 0,1,2,....

?

c x x

x

9. A coin is loaded so that heads is three times as likely as the tails. For 3 independent

tosses of the coin find the pmf of the total number of heads realized and the probability of

realizing at most 2 heads.

Suppose that the random variable X has p.d.f. given by

?

?

? ? ?

?

elsewhere

cx x

0,

, 0 1

f(x)

Find the

value of the constant c hence determine m so that

? ? 2 X m 1 P ? ?

2) Let X be a continuous random variable with pdf

?

?

? ? ? ?

?

elsewhere

k x

x

0,

, 0 3

f(x) 5

Find the value

of the constant k hence compute

P?1? X ? 3?

3) A continuous random variable Y has the pdf given by

?

?

? ? ? ?

?

elsewhere

k y x

0,

(1 ), 4 7

f(y)

Find

the value of the constant k hence compute

P?Y ? 5?

and

P?5 ? Y ? 6?

Find a linear model from the following data in the table. Express your answer in y = mx + b form. X 0 4 8 12 200 250 300 350 Solve this problem with paper/pencil first and then choose ALL that apply. There will be several answers that you need to choose from the options listed below: O The slope, m, of the linear model is - 12.50. O The slope, m, of the linear model is 12.50. O The slope, m, of the linear model is -8.33. The slope, m, of the linear model is 8.33. The slope, m, of the linear model is not given here. The y-intercept, b, of the linear model is 160.7. Consider the following two models. model A. R" 3 x - Linear(n,100) - Linear(100,8) - Linear(8,16) - softmax(16) model B. R" 3 x - Linear(n,100) - Linear(100,16) - softmax(16) (a) Is there any flaw in the design of these two models? (b) What is the advantage of model A over model B? (c) What is the advantage of model B over model A?Regression Equation in Uncoded Units RESPONSE = 5.839 - 1.345 A/C - 0.096 TR/PA + 0.0868 TI/PR + 0.400 A/C.TR/PA + 0.0189 A/C.TI/PR - 0.0200 TR/PA"TI/PR - 0.0120 A/C*TR/PA*TI/PR Regression Equation in Uncoded Units Response = 9.35 - 2.07 A/C + 0.16 TR/PA + 0.2228 TI/PR . 1.86 A/C'TR/PA . 0.0144 A/C . 0.0519 TR/PA'TV/PR . 0.0544 A/C*TR/PA*TV/PR i want an explination about the numbers in the reggression model our response is kilometer per liter we are studying the affect on the fuel our factor is AC: air conditioning TR/PA: traffic pattren TI/PR: tire pressure 2 cars GMC YOKUN and toyota corollaThe scatterplot below displays the yearly production in millions of pounds of flue-cured tobacco in the US. For what range of years is a linear model appropriate? US Flue-cured Tobacco Production 2000 1500 Production, Million Ib 1000 500 1900 1920 1940 1960 1980 2000 2020 Year O One linear model for 1919 through about 1960 and another linear model for about 1960 through 2000. O A linear model should be used for each pair of adjacent data points. O A linear model should not be used for any part of the data. O A single linear model is appropriate for the entire data set. O None of these