Advertisements
Advertisements
प्रश्न
The probability density function of the random variable X is given by
`f(x) = {{:(16x"e"^(-4x), x > 0),(0, x ≤ 0):}`
find the mean and variance of X
Advertisements
उत्तर
Given p.d.f is `f(x) = {{:(16x"e"^(-4x), x > 0),(0, x ≤ 0):}`
Mean: E(X)
= `int_-oo^oo x f(x) "d"x`
= `16int_0^oo x^2 "e"^(-4x) "d"x`
Using integration by parts method twice
Let u = x2
⇒ du = 2x dx
And `int "dv" = int "e"^(-4x)`
v = `"e"^(-4x)/(-4)`
`int "u" "d" = "uv" - int "v" "du"`
`int^2"e"^(-4x) "d"x = (x^2"e"^(-4x))/(-4) + 1/4 int 2x"e"^(-4x) "d"x` .......(1)
= `- (x^2"e"^(-4x))/4 + 1/2 int x"e"^(-4x) "d"x`
∵ Integration by parts method
u = x
⇒ du = dx
And `int "dv" - int"e"^(-4x) ""x`
v = `"e"^(-4x)/(-4)`
`int "u" "dv" = "uv" - int "v" "du"`
`int x"e"^(-4x) "d"x = (-x"e"^(-4x))/4 + 1/4 int "e"^(-4x) "d"x`
= `(-x"e"^(-4x))/4 - 1/16 "e"^(-4x)`
Substituting in (1)
`intx^2"e"^(-4x) "d"x = (x^2"e"^(-4x))/(-4) + 1/2 [(-x"e"^(-4x))/4 - 1/16 e"^(-4x)]`
E(X) = `16[(x^2"e"^(-4x))/(-4) - (x"e"^(-4x))/8 - "e"^(-4x)/32]_0^oo`
= `16[0 - ((-1)/32)]`
= `16[1/32]`
= `1/2`
E(X2] = `int_-oo^oo x^2 f(x) "d"x`
= `16int_0^oo x^3"e"^(-4x) "d"x`
Using integration by parts method
Let u = x3
⇒ du = 3x2 du
And `int "dv" = int "e"^(-4x) "d"x`
⇒ v = `"e"^(-4x)/(-4)`
`int "u" "dv" = "uv" - int "v" "du"`
`intx^3"e"^(-4x) "d"x = - (x^3"e"^(-4x))/4 + 3/4 int"e"^(-4x) x^2 "d"x`
= `- (x^3"e"^(-4x))/4 + 3/4[(x^2"e"^(-4x))/(-4) - (x"e"^(-4x))/8 - "e"^(-4x)/32]`
∵ Using E(X) integration]
= `- (x^3"e"^(-4x))/4 - 3/16 x^2"e"^(-4x) - 3/32 x"e"^(-4x) - 3/128 "e"^(-4x)`
E(X2) = `16[- (x^3""^(-4x))/4 - 3/16 x^2"e"^(-4x) - 3/32 x"e"^(-4x) - 3/128 "e"^(-4x)]_0^oo`
= `16[0 - ((-3)/128)]`
= `16[3/128]`
= `3/8`
Variance Var(X) = E(X2) – [E(X)]2
= `3/8 - 1/4`
= `(3 - 2)/8`
= `1/8`
APPEARS IN
संबंधित प्रश्न
If µ and σ2 are the mean and variance of the discrete random variable X and E(X + 3) = 10 and E(X + 3)2 = 116, find µ and σ2
Four fair coins are tossed once. Find the probability mass function, mean and variance for a number of heads that occurred
A lottery with 600 tickets gives one prize of ₹ 200, four prizes of ₹ 100, and six prizes of ₹ 50. If the ticket costs is ₹ 2, find the expected winning amount of a ticket
Choose the correct alternative:
Consider a game where the player tosses a six-sided fair die. If the face that comes up is 6, the player wins ₹ 36, otherwise he loses ₹ k2, where k is the face that comes up k = {1, 2, 3, 4, 5}. The expected amount to win at this game in ₹ is
Choose the correct alternative:
If P(X = 0) = 1 – P(X = 1). If E[X] = 3 Var(X), then P(X = 0) is
Choose the correct alternative:
If X is a binomial random variable with I expected value 6 and variance 2.4, Then P(X = 5) is
The following table is describing about the probability mass function of the random variable X
| x | 3 | 4 | 5 |
| P(x) | 0.2 | 0.3 | 0.5 |
Find the standard deviation of x.
What do you understand by Mathematical expectation?
Define Mathematical expectation in terms of discrete random variable
State the definition of Mathematical expectation using continuous random variable
Choose the correct alternative:
Value which is obtained by multiplying possible values of a random variable with a probability of occurrence and is equal to the weighted average is called
Choose the correct alternative:
Given E(X) = 5 and E(Y) = – 2, then E(X – Y) is
Choose the correct alternative:
Which of the following is not possible in probability distribution?
Choose the correct alternative:
A discrete probability distribution may be represented by
Choose the correct alternative:
A probability density function may be represented by
Choose the correct alternative:
`int_(-oo)^oo` f(x) dx is always equal to
Choose the correct alternative:
The distribution function F(x) is equal to
Prove that V(aX) = a2V(X)
