How is the variance of a random variable scaled?
@rdeyke Let’s consider a Random Variable X with mean 2 and Variance 1 (Standard Deviation also natuarally is then 1). The mean gets multiplied by the constant k, let’s say it is -2. As originally, your mean was 2, now new mean would be -2*2 = -4 Next comes the Variance. Variance is scaled by k squared.
Is the SD always the root of the variance?
The Standard Deviation is always the positive root of the Variance, and hence, the SD in this case would come out to be 2. Comment on r c’s post “@rdeyke Let’s consider a Random Variable X with me…”
How is the SD scaled for standard deviation?
Next comes the Variance. Variance is scaled by k squared. Hence, it would be multiplied by (-2)^2 which is 4. The Standard Deviation is always the positive root of the Variance, and hence, the SD in this case would come out to be 2.
What happens when you scale a density function?
Scaling a density function doesn’t affect the overall probabilities (total = 1), hence the area under the function has to stay the same one. If you multiply the random variable by 2, the distance between min (x) and max (x) will be multiplied by 2. Hence you have to scale the y-axis by 1/2.
What is the impact of scaling and shifting random?
If you multiply your x by 2 and want to keep your area constant, then x*y = 12*y = 24 => y = 24/12 = 2. Scaling the x by 2 = scaling the y by 1/2. If you didn’t scale down your y-axis, then your cumulative probabilities will be >1, which is not possible.
How does an upwards shift affect the random variable?
Because an upwards shift would imply that the probability density for all possible values of the random variable has increased (at all points). But this would consequently be increasing the area under the probability density function, which violates the rule that the area under any probability density function must be = 1 .