Question

Consider the following exponential probability density function.  for x ≥ 0 If needed, round your answer to four decimal digits.(a) Find P(x ≤ 2). (b) Find P(x ≥ 3). (c) Find P(x ≤ 5). (d) Find P(2 ≤ x ≤ 5).

Answer 1

Here the general formula is

`P(X\leq x_(0) )=1 - e^{(-x_(0))/(2) }`

a)
`P(X\leq 2 )= 0.6321`

b)
`P(X\geq 3 )= 0.2231`

c)
`P(X\leq 5 )= 0.9179`

d)
`P(2\leq X\leq 5 )= 0.2858`

Explanation:

Now the solutions are,

a)

`P(X\leq 2 )=1 - e^{(-2)/(2) }\\P(X\leq 2 )= 0.6321`

b)

`P(X\geq 3 )=1-[1 - e^{(-3)/(2) }]\\P(X\geq 3 )= 0.2231`

c)

`P(X\leq 5 )=1 - e^{(-5)/(2) }\\P(X\leq 5 )= 0.9179`

d)

`P(2\leq X\leq 5 )=1 - e^{(-5)/(2) }-1 - e^{(-2)/(2) }\\P(2\leq X\leq 5 )= 0.2858`