Advertisements
Advertisements
Question
The probability density function of X is given by
`f(x) = {{:(kx"e"^(-2x), "for" x > 0),(0, "for" x ≤ 0):}`
Find the value of k
Solution
`f(x) = {{:(kx"e"^(-2x), "for" x > 0),(0, "for" x ≤ 0):}`
Since, f(x) is a Probability density function
`int_oo^oo f(x) "d"x` = 1
i.e., `int_oo^0 0 "d"x + k int_0^oo x"e"^(-2x) "d"x` = 1
`k int_0^oo x"e"^(-2x) "d"x` = 1
Using, integration by parts method
Let u = x
`int "dv" = int "e"^(-2x) "d"x`
di = dx
v = `"e"^(-2x)/(-2)`
`int "u" "dv" = "uv" - int "v" "du"`
`int x"e"^(-2x) "d"x = (x"e"^(-2x))/(-2) + 1/2 int "e"^(-2x) "d"x`
= `- (x"e"^(-2x))/2 - "e"^(-2x)/4`
Now `k int_0^oo x"e"^(-2x) "d"x` = 1
`k[- (x"e"^(-2x))/2 - "e"^(-2x)/4]_0^oo` = 1
`k[0 + (0 + 1/4)]` = 1
⇒ `k(1/4)` = 1
k = 4
APPEARS IN
RELATED QUESTIONS
The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`
Find P(1.2 ≤ X < 1.8)
The probability density function of X is `f(x) = {(x, 0 < x < 1),(2 - x, 1 ≤ x ≤ 2),(0, "otherwise"):}`
Find P(0.5 ≤ X < 1.5)
Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the distribution function
Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 litres and a maximum of 600 litres with probability density function
`f(x) = {{:(k, 200 ≤ x ≤ 600),(0, "otherwise"):}`
Find the probability that daily sales will fall between 300 litres and 500 litres?
The probability density function of X is given by
`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`
Find the value of k
The probability density function of X is given by
`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`
Find P(X < 3)
The probability density function of X is given by
`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`
Find P(5 ≤ X)
The probability density function of X is given by
`f(x) = {(k"e"^(- x/3), "for" x > 0),(0,"for" x ≤ 0):}`
Find P(X ≤ 4)
If X is the random variable with probability density function f(x) given by,
`f(x) = {{:(x + 1",", -1 ≤ x < 0),(-x +1",", 0 ≤ x < 1),(0, "otherwise"):}`
then find the distribution function F(x)
If X is the random variable with probability density function f(x) given by,
`f(x) = {{:(x + 1",", -1 ≤ x < 0),(-x +1",", 0 ≤ x < 1),(0, "otherwise"):}`
then find P(– 0.5 ≤ x ≤ 0.5)
If X is the random variable with distribution function F(x) given by,
F(x) = `{{:(0",", - oo < x < 0),(1/2(x^2 + x)",", 0 ≤ x ≤ 1),(1",", 1 ≤ x < oo):}`
then find the probability density function f(x)
Choose the correct alternative:
Let X be random variable with probability density function
`f(x) = {(2/x^3, x ≥ 1),(0, x < 1):}`
Which of the following statement is correct?
Choose the correct alternative:
The random variable X has the probability density function
`f(x) = {{:("a"x + "b", 0 < x < 1),(0, "otherwise"):}`
and E(X) = `7/12`, then a and b are respectively
Choose the correct alternative:
If `f(x) = {{:(2x, 0 ≤ x ≤ "a"),(0, "otherwise"):}` is a probability density function of a random variable, then the value of a is
Choose the correct alternative:
A computer salesperson knows from his past experience that he sells computers to one in every twenty customers who enter the showroom. What is the probability that he will sell a computer to exactly two of the next three customers?
