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A randomm variable X takes the value -1,...

A randomm variable X takes the value -1,0,1. It's mean is 0.6. If (X=0)=0.2 then P(X=1)=

A

0.3

B

0.5

C

0.7

D

0.8

Text Solution

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The correct Answer is:
To solve the problem, we need to find the probability \( P(X=1) \) given the conditions of the random variable \( X \). ### Step-by-step Solution: 1. **Define the Probabilities**: Let: - \( P(X = -1) = c \) - \( P(X = 0) = b = 0.2 \) - \( P(X = 1) = a \) 2. **Use the Total Probability**: Since the total probability must sum to 1, we have: \[ P(X = -1) + P(X = 0) + P(X = 1) = 1 \] Substituting the known values: \[ c + 0.2 + a = 1 \] This simplifies to: \[ a + c = 0.8 \quad \text{(Equation 1)} \] 3. **Use the Mean**: The mean of the random variable \( X \) is given as 0.6. The mean can be calculated as: \[ \text{Mean} = \sum (x \cdot P(X = x)) = (-1) \cdot c + 0 \cdot b + 1 \cdot a \] Substituting the known values: \[ -c + 0 + a = 0.6 \] This simplifies to: \[ a - c = 0.6 \quad \text{(Equation 2)} \] 4. **Solve the System of Equations**: Now we have two equations: - Equation 1: \( a + c = 0.8 \) - Equation 2: \( a - c = 0.6 \) We can add these two equations: \[ (a + c) + (a - c) = 0.8 + 0.6 \] This simplifies to: \[ 2a = 1.4 \implies a = 0.7 \] 5. **Find \( c \)**: Now substitute \( a = 0.7 \) back into Equation 1: \[ 0.7 + c = 0.8 \implies c = 0.8 - 0.7 = 0.1 \] 6. **Conclusion**: We have found: - \( P(X = 1) = a = 0.7 \) - \( P(X = -1) = c = 0.1 \) - \( P(X = 0) = b = 0.2 \) Thus, the probability \( P(X = 1) \) is \( \boxed{0.7} \).
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