Ex. 1. Find the probability distribution of the number of heads when a fair coin is tossed repeatedly until the first tail appears.
Ex. 2. A random variable X has the following probability mass function
| X: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
| P(X = k) = f(x): |
0 |
k |
2k |
2k |
3k |
k2 |
2k2 |
7k2+k |
(i) determine the constant k.
(ii) evaluate P(X<6), P(X≥6), P(3
(iii) find the minimum value of x so that P(X≤x)>1/2
(iv) obtain the distribution function F(x)
Ex.3. Let F(x) = 0 -∞ < x < 1
= 1/5 1 ≤ x ≤ 2
= 3/5 1 ≤ x < 3
= 1 3 ≤ x < ∞
Show that F(x) is a possible distribution function. Determine the spectrum and the probability mass of the distribution.
Ex. 4. The distribution function F(x) of a variate X is defined as follows
F(x) = A - ∞ < x < -1
= B -1 ≤ x < 0
= C 0 ≤ x < 2
= D 2 ≤ x < ∞
Where A,B,C,D are constants. Determine the values of A, B, C, D given that P( X=0)= 1/6 and P( X>1 )= 2/3.