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The co-efficient of correlation between two variables x and y is 0.5, their covariance is 16. If the S.D. of x is 4, then the S.D. of y is equal to
- a)4
- b)8
- c)16
- d)64
Correct answer is option 'B'. Can you explain this answer?
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The co-efficient of correlation between two variables x and y is 0.5, ...
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The co-efficient of correlation between two variables x and y is 0.5, ...
Given: Coefficient of correlation between x and y = 0.5
Covariance of x and y = 16
Standard deviation of x = 4
To find: Standard deviation of y
Formulae used:
Covariance of x and y = E[(x - mean_x) * (y - mean_y)]
Coefficient of correlation between x and y = Covariance of x and y / (Standard deviation of x * Standard deviation of y)
Standard deviation of y = sqrt(Var(y))
Solution:
1. Finding the mean of x and y:
Mean of x = mean_x = 0 (since x is centered at 0, as given in the question)
Mean of y = mean_y = E(y) = E[(0.5 * x) + u] = 0.5 * E(x) + E(u) = 0 (since E(x) = 0 and E(u) = 0)
2. Finding the variance and standard deviation of x:
Variance of x = Var(x) = E[(x - mean_x)^2] = E[x^2] = (Standard deviation of x)^2 = 16
Therefore, Standard deviation of x = sqrt(16) = 4
3. Finding the covariance of x and y:
Covariance of x and y = E[(x - mean_x) * (y - mean_y)] = E[(x - 0) * (0.5 * x - 0)] = 0.5 * E[x^2] = 0.5 * Var(x) = 0.5 * 16 = 8
4. Finding the standard deviation of y:
Coefficient of correlation between x and y = Covariance of x and y / (Standard deviation of x * Standard deviation of y)
0.5 = 8 / (4 * Standard deviation of y)
Standard deviation of y = 8 / (0.5 * 4) = 8
Therefore, the standard deviation of y is 8, which is option (B).
Covariance of x and y = 16
Standard deviation of x = 4
To find: Standard deviation of y
Formulae used:
Covariance of x and y = E[(x - mean_x) * (y - mean_y)]
Coefficient of correlation between x and y = Covariance of x and y / (Standard deviation of x * Standard deviation of y)
Standard deviation of y = sqrt(Var(y))
Solution:
1. Finding the mean of x and y:
Mean of x = mean_x = 0 (since x is centered at 0, as given in the question)
Mean of y = mean_y = E(y) = E[(0.5 * x) + u] = 0.5 * E(x) + E(u) = 0 (since E(x) = 0 and E(u) = 0)
2. Finding the variance and standard deviation of x:
Variance of x = Var(x) = E[(x - mean_x)^2] = E[x^2] = (Standard deviation of x)^2 = 16
Therefore, Standard deviation of x = sqrt(16) = 4
3. Finding the covariance of x and y:
Covariance of x and y = E[(x - mean_x) * (y - mean_y)] = E[(x - 0) * (0.5 * x - 0)] = 0.5 * E[x^2] = 0.5 * Var(x) = 0.5 * 16 = 8
4. Finding the standard deviation of y:
Coefficient of correlation between x and y = Covariance of x and y / (Standard deviation of x * Standard deviation of y)
0.5 = 8 / (4 * Standard deviation of y)
Standard deviation of y = 8 / (0.5 * 4) = 8
Therefore, the standard deviation of y is 8, which is option (B).
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The co-efficient of correlation between two variables x and y is 0.5, their covariance is 16. If the S.D. of x is 4, then the S.D. of y is equal toa)4b)8c)16d)64Correct answer is option 'B'. Can you explain this answer?
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