Final answer:
To calculate the mean battery lifetime, integrate the PDF over the entire range and multiply by X. To find the standard deviation, calculate the variance first. The mode is the value with the highest point on the PDF. The CDF gives the probability that the battery lifetime is less than or equal to a certain value.
Step-by-step explanation:
The battery lifetime of a cell phone is modeled by a random variable X with a probability density function (PDF). To calculate the mean battery lifetime, you need to integrate the PDF over the entire range of X and multiply by X:
Mean battery lifetime = Integral(X * PDF(X)) dX
To determine the standard deviation, you need to calculate the variance first by integrating (X - Mean)^2 * PDF(X) over the entire range of X:
Variance = Integral((X - Mean)^2 * PDF(X)) dX
The mode of the battery lifetime is the value of X that corresponds to the highest point on the PDF. To find it, you can take the derivative of the PDF and set it to zero, then solve for X:
PDF'(X) = 0, solve for X
The cumulative distribution function (CDF) gives the probability that the battery lifetime is less than or equal to a certain value. To evaluate the CDF, integrate the PDF from negative infinity to X:
CDF(X) = Integral(PDF(t)) dt, from negative infinity to X