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) = {{:(1/10, x = 2"," 5),(1/5, x = 0"," 1"," 3"," 4):}`
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
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
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
Find the expected value for the random variable of an unbiased die
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
In a business venture a man can make a profit of ₹ 2,000 with a probability of 0.4 or have a loss of ₹ 1,000 with a probability of 0.6. What is his expected, variance and standard deviation of profit?
Choose the correct alternative:
Probability which explains x is equal to or less than a particular value is classified as
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:
Which of the following is not possible in probability distribution?
Choose the correct alternative:
A listing of all the outcomes of an experiment and the probability associated with each outcome is called
Choose the correct alternative:
An expected value of a random variable is equal to it’s
Choose the correct alternative:
A discrete probability function p(x) is always non-negative and always lies between
Prove that V(aX) = a2V(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
