1-Let X be a discrete random variable with a probability function given byp(x) = c

7

x, para x = 1, . . . , 6 where is one

positive real constant. Determine:

(a) the value of the constant c.

(b) the probability that X is an odd number.

(c) the expected value of X.

(d) the variance of X

2-Let X be a discrete random variable with a distribution functionF(x)0 , para x < 0;

1

2

, para 0 x < 1;

3

5

, para 1 x < 2;

4

5

, para 2 x < 3;

9

10 , para 3 x < 4;

1 , para x 4

a) The probability distribution of X;

(b) F (2) and F (2);

(c) P (0, 5 X 2, 5);

(d) P (X = 0 | X is even).

3-Let X be a continuous random variable with a distribution function F(x) = 0 , para x < 0;

x^2/2, para 0 x < 1/2

x^3, para 1/2 x < 1;

1 , para x 1

Determine:

(a) the probability density function of X; (b) Calculate E (X);

(c) Calculate the expected value Y = 2X - 1; (d) Calculate P (1/4 X 3/4 | X >= 1/2