Expectations
Question
Q1.
Expectation of a constant is
(a) one (b) zero (c) two (d) constant
Solution.
(d) constant
E(a) = a
Solution
Q2.
A random variable
(a) Always positive
(b) Always negative
(c) Non-zero
(d) May be real
(d) May be real
Solution
Q3.
Find E(4x + 3)2 , if p.d.f is
| x |
0 |
1 |
2 |
3 |
| p(x) |
1/8 |
3/8 |
3/8 |
1/8 |
given E(x) = 1.5 and E(x2) = 3
Solution.
E(4x+3)2 = E(16x2 + 24x + 9)
= E(16x2) + E(24x) + E(9)
= 16 E(x2) + 24 E(x) + E(9)
16 x 3 + 24 x (1.5) + 9
= 93
Solution
Q4.
Find v(x) and v(4x+9) if p.d.f
| x |
0 |
1 |
2 |
3 |
| p(x) |
1/8 |
3/8 |
3/8 |
1/8 |
E(x) = 1.5 and E(x2) = 3
Solution.
v(x) = E(x2) - {E(x)}2 = 3 - 2.25 = 0.75
v(4x+9) = 16v(x) = 16 x 0.75 = 12
Solution
Q5.
If E(4) = 4, v(4) = ___________
Solution.
v(4) = 0
∴ v(a) = 0 {a = any constant}
Solution
Q6.
A bag contains 3 red and 2 black balls . A person selects two balls at random find the expected number of black balls .
Solution.
Let x = is the random variable of black balls
x = 0,1,2 .
| No. of black balls x |
p(x) |
x p(x) |
| 0 |
2C0 x 3C2/5C2 =
3/10 |
0 |
| 1 |
2C1 x 3C1/5C2 =
6/10 |
6/10 |
| 2 |
2C2 x 3C0/5C2 =
1/10 |
2/10 |
E(x) = ∑xP(x) = 0 +
6/10
+
2/10
=
8/10
= 0.8
Solution
Q7.
Random variable can be classified .
(a) Discrete (b) Continuous (c) Both (d) None of these
Solution.
(c) Both .
Solution
Q8.
Expectation of a deviation , from mean is
(a) one (b) two (c) zero (d) three
Solution.
(c) zero
Solution
Q9.
An urn contains 4 white and 2 black balls.A person selects 3 balls at random and he receives Rs. 5 for each black balls and has to pay Rs.3 for each
white ball . Find E(x).
Solution.
In this case, the random variable is the amount gained by the person .
| No. of black balls x |
Amount gained |
p(xi) |
x . p(x) |
| 0 |
-9 |
4C3 x 2C0/6C3 =
4/20 |
- 36/20 |
| 1 |
-1 |
4C2 x 2C1/6C3 =
12/20 |
- 12/20 |
| 2 |
7 |
4C1 x 2C2/6C3 =
4/20 |
28/20 |
E(x) = -
36/20
-
12/20
+
28/20
=
-48/20
-
28/20
=
-20/20
E(x) = -1
Note : 0 black ball = 3(-3) = -9
1 black, 2 white = 1x5+2(-3) = -1
2 black, 1 white = 2x5+1(-3) = 7
Solution
Q10.
E(x.y) = ____________
(a) E(x)-E(4) (b) E(x)+E(x) (c) E(x).E(y) (d)
E(x)/E(y)
Solution.
(c) E(x).E(y)
Solution
