A drawer contains 8 brown socks and 4 blue socks. A sock is taken from the drawer at random, its colour is noted and it is then replaced. This procedure perform twice more. If X is the random variable. “the number of brown socks taken “,
a) Construct the probability distribution table (5M)
b) Compute E(X) (6M)
c) Calculate E (5X +3) (4M)
d) Calculate VAR (X) (5M)
Please solve this question
<p>Test</p>
a) Probability distribution table:
X = number of brown socks taken
X | P(X)
---------------
0 | 1/12
1 | 1/4
2 | 1/4
3 | 1/4
b) E(X) = Σ(X * P(X))
E(X) = (0 * 1/12) + (1 * 1/4) + (2 * 1/4) + (3 * 1/4)
E(X) = 0 + 1/4 + 1/2 + 3/4
E(X) = 2
c) E(5X + 3) = 5 * E(X) + 3 * 1
E(5X + 3) = 5 * 2 + 3
E(5X + 3) = 10 + 3
E(5X + 3) = 13
d) VAR(X) = E(X^2) - [E(X)]^2
To calculate E(X^2), we need to find the sum of the squares of each value of X multiplied by its corresponding probability:
E(X^2) = (0^2 * 1/12) + (1^2 * 1/4) + (2^2 * 1/4) + (3^2 * 1/4)
E(X^2) = (0 + 1 + 4 + 9) / 4
E(X^2) = 14/4
E(X^2) = 3.5
Now we can calculate VAR(X):
VAR(X) = E(X^2) - [E(X)]^2
VAR(X) = 3.5 - (2)^2
VAR(X) = 3.5 - 4
VAR(X) = -0.5
Note: The variance cannot be negative, so a possible error may have occurred during the calculations. Please double-check the calculations or the given information to ensure accuracy.
To answer these questions, we need to calculate the probabilities of picking a brown or blue sock at each draw and then use those probabilities to construct the probability distribution table.
a) Probability distribution table:
To construct the probability distribution table, we need to calculate the probabilities of each possible outcome.
Let's define the events:
B = Drawing a brown sock
C = Drawing a blue sock
1st draw:
P(B) = 8/12 (since there are 8 brown socks out of 12 total socks)
P(C) = 4/12 (since there are 4 blue socks out of 12 total socks)
2nd draw:
P(BB) = P(B) * P(B) = (8/12) * (8/12) = 64/144
P(BC) = P(B) * P(C) = (8/12) * (4/12) = 32/144
P(CB) = P(C) * P(B) = (4/12) * (8/12) = 32/144
P(CC) = P(C) * P(C) = (4/12) * (4/12) = 16/144
3rd draw:
P(BBB) = P(BB) * P(B) = (64/144) * (8/12) = 512/1728
P(BBC) = P(BB) * P(C) = (64/144) * (4/12) = 256/1728
P(BCB) = P(BC) * P(B) = (32/144) * (8/12) = 256/1728
P(BCC) = P(BC) * P(C) = (32/144) * (4/12) = 128/1728
P(CBB) = P(CB) * P(B) = (32/144) * (8/12) = 256/1728
P(CBC) = P(CB) * P(C) = (32/144) * (4/12) = 128/1728
P(CCB) = P(CC) * P(B) = (16/144) * (8/12) = 128/1728
P(CCC) = P(CC) * P(C) = (16/144) * (4/12) = 64/1728
Now we can construct the probability distribution table:
| X | Probability |
|-------|----------------|
| 0 | 1/27 |
| 1 | 8/27 |
| 2 | 12/27 |
| 3 | 6/27 |
b) Expected value of X (E(X)):
The expected value of a random variable is calculated by multiplying each possible value of the variable by its corresponding probability and summing up the results.
E(X) = (0 * 1/27) + (1 * 8/27) + (2 * 12/27) + (3 * 6/27)
= 0 + (8/27) + (24/27) + (18/27)
= 50/27
≈ 1.85
So, E(X) is approximately 1.85.
c) Expected value of (5X + 3) (E(5X + 3)):
To calculate the expected value of (5X + 3), we need to multiply each possible value of X by 5, add 3, and then calculate the expected value as we did before.
E(5X + 3) = (5 * 0 + 3) * (1/27) + (5 * 1 + 3) * (8/27) + (5 * 2 + 3) * (12/27) + (5 * 3 + 3) * (6/27)
= (3 * 1/27) + (8 * 8/27) + (13 * 12/27) + (18 * 6/27)
= 126/27
= 4.67
So, E(5X + 3) is approximately 4.67.
d) Variance of X (VAR(X)):
The variance of a random variable measures how spread out the possible values are.
VAR(X) = E(X^2) - (E(X))^2
First, let's calculate E(X^2) by multiplying each possible value of X squared by its corresponding probability and summing up the results.
E(X^2) = (0^2 * 1/27) + (1^2 * 8/27) + (2^2 * 12/27) + (3^2 * 6/27)
= 0 + (8/27) + (48/27) + (54/27)
= 110/27
≈ 4.07
Now, we can calculate the variance:
VAR(X) = E(X^2) - (E(X))^2
= 110/27 - (50/27)^2
= 110/27 - 2500/729
= 110/27 - 2500/729 * 27/27
= 110/27 - 2500/729 * 27/27
= 110/27 - 2500/729
≈ 0.74
So, VAR(X) is approximately 0.74.
<table>
X: variable random
So 8 brown socks plus 4 blue socks in the drawer=12 socks
Let assume that P(no brown socks) means P(X=0)=(4/12)³ must be equal to 1/27
P(X=1)=[8/12(4/12)² +(4/12)²8/12 +(4/12×8/12×4/12)]=6/27
P(X=2)= 3(8/12×8/12×4/12) =4/9
P(X=3) =3(8/12)³ =8/27