pease find solutions

On a disc with origin (0, 0) and radius 1, a point (X, Y ) is selected by

throwing a dart that hits the disc in an arbitrary place. This is best described

by the joint probability density function f of X and Y , given by

f(x, y) =

c if x2 + y2 ? 1

0 otherwise,

where c is some positive constant.

a. Determine c.

b. Let R = ?

X2 + Y 2 be the distance from (X, Y ) to the origin. Determine

the distribution function FR.

c. Determine the marginal density function fX. Without doing any calculations, what can you say about fY ?

9.14 An arbitrary point (X, Y ) is drawn from the square [?1, 1] [?1, 1].

This means that for any region G in the plane, the probability that (X, Y ) is

in G, is given by the area of G? divided by the area of , where denotes

the square [?1, 1] [?1, 1]:

P((X, Y ) ? G) = area of G ?

area of .

a. Determine the joint probability density function of the pair (X, Y ).

b. Check that X and Y are two independent, U(?1, 1) distributed random

variables.

9.15 Let the pair (X, Y ) be drawn arbitrarily from the triangle ? with

vertices (0, 0), (0, 1), and (1, 1).

a. Use Figure 9.5 to show that the joint distribution function F of the pair

(X, Y ) satisfies

F(a, b) =

?

???????

???????

0 for a or b less than 0

a(2b ? a) for (a, b) in the triangle ?

b2 for b between 0 and 1 and a larger than b

2a ? a2 for a between 0 and 1 and b larger than 1

1 for a and b larger than 1.

b. Determine the joint probability density function f of the pair (X, Y ).

c. Show that fX(x)=2 ? 2x for x between 0 and 1 and that fY (y)=2y for

y between 0 and 1.

How many chords can be drawn through 21 points on a circle?

4. How many triangles can be formed by joining the vertices of a hexagon?

5. Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels

can be formed?

6. If four dice are rolled, find the number of possible outcomes in which atleast one

die shows 2.

7. There are 18 guests at a dinner party. They have to sit 9 guests on either side of

a long table, three particular persons decide to sit on one particular side and two

others on the other side. In how many ways can the guests to be seated?

8. If a polygon has 44 diagonals, find the number of its sides.

9. In how many ways can a cricket team of 11 players be chosen out of a batch of 15

players?

(i) There is no restriction on the selection.

(ii) A particular player is always chosen.

(iii) A particular player is never chosen.

10. A Committee of 5 is to be formed out of 6 gents and 4 ladies. In how many ways this

can be done when

(i) atleast two ladies are included

(ii) atmost two ladies are include

QUESTION 2: Linear transformation of random variables (10) a. If X is a normally distributed random variable with mean of 3 and variance of 2, what is the mean and variance of a transform random variable Y, where Y = 5X + 8? b. More generally, what is the mean and variance of a random variable Y = ax + b, where X is itself a random variable with a mean of u and variance of o ? Write your answer in terms of a, b, u, and g2.Suppose we take independent samples of sizes n = 8 and no =6 from two normal populations with equal population variances, i.e. of = of a) Use R to find the following: al) P($1/S; > 3) a.2) P($1/S;