Financial Risk Manager Part 1
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Given a binomial random variable with E(X) = 2 and Var(X) = 1.5, calculate P(X=5).
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A
0.015
B
0.023
C
0.039
D
0.047
Explanation:
Explanation
For a binomial distribution:
- Mean (E(X)) = np = 2
- Variance (Var(X)) = np(1-p) = 1.5
Step 1: Find p and n
From the variance formula:
Var(X) = np(1-p) = 1.5
2(1-p) = 1.5 (since np = 2)
1-p = 1.5/2 = 0.75
p = 1 - 0.75 = 0.25
Var(X) = np(1-p) = 1.5
2(1-p) = 1.5 (since np = 2)
1-p = 1.5/2 = 0.75
p = 1 - 0.75 = 0.25
Now find n:
np = 2
n × 0.25 = 2
n = 2/0.25 = 8
np = 2
n × 0.25 = 2
n = 2/0.25 = 8
Step 2: Calculate P(X=5)
Using the binomial probability formula:
P(X = 5) = C(8,5) × (0.25)^5 × (0.75)^3
P(X = 5) = C(8,5) × (0.25)^5 × (0.75)^3
Where:
- C(8,5) = 56 (combinations)
- (0.25)^5 = 0.0009765625
- (0.75)^3 = 0.421875
Calculation:
P(X = 5) = 56 × 0.0009765625 × 0.421875
= 56 × 0.0004119873046875
= 0.0230712890625 ≈ 0.023
P(X = 5) = 56 × 0.0009765625 × 0.421875
= 56 × 0.0004119873046875
= 0.0230712890625 ≈ 0.023
Therefore, P(X=5) = 0.023, which corresponds to option B.
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