Advertisements
Advertisements
प्रश्न
For the random variable X with the given probability mass function as below, find the mean and variance.
`f(x) = {{:(1/2 "e"^(x/2), "for" x > 0),(0, "otherwise"):}`
Advertisements
उत्तर
Mean: `mu = "E"("X")`
= `int_0^oo x f(x) "d"x`
= `int_0^oo x 1/2 "e"^((-x)/2) "d"x`
= `1/2 [- 2x"e"^((-x)/2) ]_0^oo + int_0^oo 2"e"^((-x)/2) "d"x`
= `1/2[- 2x"e"^((-x)/2) + (2"e"^((-x)/2))/(- 1/2)]_0^oo`
= `1/2 [- 2x"e"^((-x)/2) - 4"e"^((-x)/2)]_0^oo`
= `1/2 [0 - 0 - (0 - 4)]`
= 2
Variance: `"E"("X"^2)`
= `int_0^oo x^2 f(x) "d"x`
`int u "dv" = "uv" - int "v" "du"`
u = `x int "dv" = int "e"^((-x)/2) "d"x`
du =dx, v = `"e"^((-x)/2)/(- 1/2)`
v = `- 2"e"^((-x)/2)`
Bernoulli's formula
`int"u" "dv" = "uv" - "u'v"_1 + "u''v"_2 - ......`
u = x2, `int"dv" = - int"e"^((-x)/2) "d"x`
u' = 2x, v = `- 2"e"^((-x)/2)`
u'' = 2, v1 = `- 4"e"^((-x)/2)`
u'' = 0, v2 = `- 8"e"^((-x)/2)`
`"E"("X"^2) = 1/2 int_0^oo x^2"e"^((-x)/2) "d"x`
= `1/2[-2x^2 "e"^((-x)/2) - 8x"e"^((-x)/2) - 16"e"^((-x)/2)]_0^oo`
= `1/2 [0 - (- 16"e"^0)]`
= `1/2 xx 16`
= 8
Var(X) = E(X2) – [E(X)]2
= 8 – 4
= 4
APPEARS IN
संबंधित प्रश्न
For the random variable X with the given probability mass function as below, find the mean and variance.
`f(x) = {{:(2(x - 1), 1 < x ≤ 2),(0, "otherwise"):}`
Two balls are drawn in succession without replacement from an urn containing four red balls and three black balls. Let X be the possible outcomes drawing red balls. Find the probability mass function and mean for X
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:
Four buses carrying 160 students from the same school arrive at a football stadium. The buses carry, respectively, 42, 36, 34, and 48 students. One of the students is randomly selected. Let X denote the number of students that were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on that bus. Then E(X) and E(Y) respectively are
Find the expected value for the random variable of an unbiased die
Let X be a continuous random variable with probability density function
`"f"_x(x) = {{:(2x",", 0 ≤ x ≤ 1),(0",", "otherwise"):}`
Find the expected value of X
What are the properties of Mathematical expectation?
What do you understand by Mathematical expectation?
Choose the correct alternative:
If X is a discrete random variable and p(x) is the probability of X, then the expected value of this random variable is equal to
Choose the correct alternative:
E[X – E(X)] is equal to
Choose the correct alternative:
E[X – E(X)]2 is
Choose the correct alternative:
`int_(-oo)^oo` f(x) dx is always equal to
Prove that if E(X) = 0, then V(X) = E(X2)
Prove that V(aX) = a2V(X)
Prove that V(X + b) = V(X)
The time to failure in thousands of hours of an important piece of electronic equipment used in a manufactured DVD player has the density function
f(x) = `{{:(2"e"^(-2x)",", x > 0),(0",", "otherwise"):}`
Find the expected life of this piece of equipment
