Let X be an exponential random variable and let Y=aX+b . The random variable Y is exponential if and only if (choose one of the following statements):
a>0 and b!= 0
b=0
Y is exponential if and only if every time Y tells a joke, the laughter follows an exponential distribution among the audience. And if I were to choose one of the following statements, I would pick the one that says "Y is exponential if and only if the rate parameter of Y is equal to the rate parameter of X." Make sure Y's comedy timing is on point!
The random variable Y is exponential if and only if the parameters a and b are such that a>0 and b≥0 .
To determine whether the random variable Y = aX + b is exponential, we need to consider the properties of exponential random variables.
An exponential random variable X with parameter λ has the probability density function (pdf) given by:
fX(x) = λ * e^(-λx) for x >= 0
To determine if Y = aX + b is exponential, we need to find the probability density function (pdf) of Y and check if it meets the criteria for an exponential distribution.
1. Calculate the cumulative distribution function (CDF) of Y:
FY(y) = P(Y <= y) = P(aX + b <= y)
2. Substitute X = (Y - b) / a into the equation:
FY(y) = P(X <= (y - b) / a) = FX((y - b) / a)
3. Differentiate FY(y) to get the pdf of Y:
fy(y) = d(FY(y)) / dy
If fy(y) follows the form of an exponential distribution, then Y is exponential.
So, the correct statement should be:
"The random variable Y is exponential if and only if the pdf of Y, fy(y), follows the form of an exponential distribution."
a>0
b=0