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A random variable ‘X’ is uniformly distributed in the interval [-2, 3] . Another random variable Y is related to random variable X as Y =2X+3. The value of variance of Y is
- a)5.21
- b)6.33
- c)8.33
- d)10.51
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
A random variable ‘X’ is uniformly distributed in the interval [-2, 3...
Given that random variable is uniformly distributed in the interval[-2, 3] .
PDF of X is as shown below,
Given, Y=2X+3
Var (Y)= Var (2X)+Var(3)
Var (Y) =22 Var (X)+0
Var (Y) = 4Var (X)
Var (Y) =4*25/12=8.33
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A random variable ‘X’ is uniformly distributed in the interval [-2, 3...
Understanding the Problem
The random variable X is uniformly distributed in the interval [-2, 3]. We need to find the variance of Y, which is defined as Y = 2X + 3.
Step 1: Variance of X
- For a uniform distribution over the interval [a, b], the variance is given by the formula:
Variance(X) = (b - a)² / 12.
- Here, a = -2 and b = 3.
- Thus,
Variance(X) = (3 - (-2))² / 12 = (5)² / 12 = 25 / 12 = 2.0833.
Step 2: Transformation of the Random Variable
- The random variable Y is a linear transformation of X:
Y = 2X + 3.
- In linear transformations, the variance is affected only by the coefficient of X. The constant (3) does not affect the variance.
Step 3: Calculating the Variance of Y
- The formula for the variance of a transformed variable is:
Variance(Y) = (coefficient of X)² × Variance(X).
- Here, the coefficient is 2:
Variance(Y) = (2)² × Variance(X) = 4 × (25 / 12).
- Simplifying this gives:
Variance(Y) = 100 / 12 = 8.3333.
Conclusion
- Therefore, the variance of Y is approximately 8.33, which corresponds to option 'C'.
The random variable X is uniformly distributed in the interval [-2, 3]. We need to find the variance of Y, which is defined as Y = 2X + 3.
Step 1: Variance of X
- For a uniform distribution over the interval [a, b], the variance is given by the formula:
Variance(X) = (b - a)² / 12.
- Here, a = -2 and b = 3.
- Thus,
Variance(X) = (3 - (-2))² / 12 = (5)² / 12 = 25 / 12 = 2.0833.
Step 2: Transformation of the Random Variable
- The random variable Y is a linear transformation of X:
Y = 2X + 3.
- In linear transformations, the variance is affected only by the coefficient of X. The constant (3) does not affect the variance.
Step 3: Calculating the Variance of Y
- The formula for the variance of a transformed variable is:
Variance(Y) = (coefficient of X)² × Variance(X).
- Here, the coefficient is 2:
Variance(Y) = (2)² × Variance(X) = 4 × (25 / 12).
- Simplifying this gives:
Variance(Y) = 100 / 12 = 8.3333.
Conclusion
- Therefore, the variance of Y is approximately 8.33, which corresponds to option 'C'.
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A random variable ‘X’ is uniformly distributed in the interval [-2, 3] . Another random variable Y is related to random variable X as Y =2X+3. The value of variance of Y isa)5.21b)6.33c)8.33d)10.51Correct answer is option 'C'. Can you explain this answer?
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