Question

In words, x=. the theoretical distribution of X is X (squiggly line) Exp (2) in theory, based upon the distribution X (squiggly line) Exp (2), what is the mean, the standard deviation, the first quartile, the third quartile, and the median? (please write large enough so I can see on screen)

Answer 1

X ~ Exp( = 2)

Mean = 1/ = 1/2

Standard deviation = 1/ = 1/2

The CDF of X is,

F(X) = 1 - exp(-x) = 1 - exp(-2x)

For first quartile, F(X) = 0.25

1 - exp(-2x) = 0.25

exp(-2x) = 0.75

-2x = ln(0.75)

x = 0.2876821 / 2 = 0.1438

So, the first quartile is 0.1438

For third quartile, F(X) = 0.75

1 - exp(-2x) = 0.75

exp(-2x) = 0.25

-2x = ln(0.25)

x = 1.386294 / 2 = 0.693147

So, the third quartile is 0.6931

For median, F(X) = 0.5

1 - exp(-2x) = 0.5

exp(-2x) = 0.5

-2x = ln(0.5)

x = 0.6931472 / 2 = 0.3465736

So, the third quartile is 0.3466