Question

# Find the expected values and variances of the exponential random variables with the density functions given below \frac{1}{4}e^{-\frac{x}{4}}

Random variables
Find the expected values and variances of the exponential random variables with the density functions given below
$$\frac{1}{4}e^{-\frac{x}{4}}$$

2021-06-04
Step 1
it is given that x is a exponential random variables with pdf,
$$f(x)=\frac{1}{4}e^{-\frac{x}{4}}$$
Step 2
Pdf of exponential distribution
$$f(x)=0e^{-0x}, x\geq 0$$
Mean of exponential distribution $$= \frac{1}{0}$$
Variance of exponential distribution $$= \frac{1}{0^{2}}$$
Given pdf, $$f(x)=\frac{1}{4}e^{-\frac{x}{4}}$$
Comparing it with pdf of exponential distribution, $$0 =\frac{1}{4}$$
Mean of exponential distribution $$= \frac{1}{\frac{1}{4}}=4$$
Variance of exponential distribution $$= \frac{1}{(\frac{1}{4})^{2}}=16$$