Since X is a random variable.
Assuming the random variable to be discrete..
The given probability distribution would be a discrete probability distribution of "X"
For a discrete probability distribution of a random variable "X", Σ p = 1
From the above distribution
Σ p = 1
⇒ 0 + 2K + 2K + K + 3K + K2 + 2K2 + (7K2 + K) = 1
⇒ 9k + 10 k2 = 1
⇒ 10 K2 + 9K − 1 = 0
⇒ 10 K2 + 10K − k − 1 = 0
⇒ 10 K (K + 1) − 1 (K + 1) = 0
⇒ (10K − 1) (K + 1) = 0
⇒ 10K − 1 = 0 (Or) K + 1 = 0
⇒ 10K = 1 (Or) K = − 1
Since probability cannot be negative, k = −1 is ignored.
The probability distribution replacing the values of of "k" would be
|
x |
P (X = x) |
|
In K terms |
Calculations |
Probability |
|
0 |
0 |
|
0 |
|
1 |
2k |
2 × (0.1) |
0.2 |
|
2 |
2k |
2 × (0.1) |
0.2 |
|
3 |
k |
1 × (0.1) |
0.1 |
|
4 |
3k |
3 × (0.1) |
0.3 |
|
5 |
k2 |
(0.1)2 |
0.01 |
|
6 |
2k2 |
2 × (0.1) |
0.02 |
|
7 |
7k2 + k |
7 × (0.1) + (0.1) |
0.17 |
Therefore,
P(x < 6) |
= |
P(x=0) + P (x = 1) + P (x = 2) + P (x = 3) + P (x = 4) + P (x = 5) |
(Or) |
= |
1 − P(x ≥ 6) |
|
= |
1 − [P(x=6) + P(x=7)] |
|
= |
1 − [0.02 + 0.17] |
|
= |
1 − 0.19 |
|
= |
0.81 |
P(x ≥ 6) |
= |
P(x=6) + P(x=7) |
|
= |
0.02 + 0.17 |
|
= |
0.19 |
P(0 < x < 5) |
= |
P(x=1) + P(x=2) + P(x=3) + P(x=4) |
|
= |
0.2 + 0.2 + 0.1 + 0.3 |
|
= |
0.8 |
Calculations for finding the mean and variance of the distribution
|
x |
P (X = x) |
px [x × P (X = x)] |
x2 |
px2 [x2 × P (X = x)] |
|
0 |
0.02 |
0 |
0 |
0 |
|
1 |
0.20 |
0.20 |
1 |
0.20 |
|
2 |
0.20 |
0.40 |
4 |
0.80 |
|
3 |
0.10 |
0.30 |
9 |
0.90 |
|
4 |
0.30 |
1.20 |
16 |
4.80 |
|
5 |
0.01 |
0.05 |
25 |
0.25 |
|
6 |
0.02 |
0.12 |
36 |
0.72 |
|
7 |
0.17 |
1.19 |
49 |
8.33 |
Total |
|
1.00 |
3.46 |
|
16.00 |
Mean/Expectation of the distribution
⇒ E (x) (Or) x |
= |
Σ px |
|
= |
3.46 |
Variance of the distribution
⇒ var (x) |
= |
E (x2) − (E(x))2 |
⇒ var (x) |
= |
Σ px2 − (Σ px)2 |
|
= |
16 − (3.46)2 |
|
= |
16 − 11.9716 |
|
= |
4.0284 |
Standard Deviation of the distribution
⇒ SD (x) |
= |
+ √ Var (x) |
⇒ SD (x) |
= |
+ √ 4.0284 |
⇒ SD (x) |
= |
+ 2.007 |