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
Find the expected value for the random variable of an unbiased die
Let X be a random variable defining number of students getting A grade. Find the expected value of X from the given table:
| X = x | 0 | 1 | 2 | 3 |
| P(X = x) | 0.2 | 0.1 | 0.4 | 0.3 |
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.
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
In investment, a man can make a profit of ₹ 5,000 with a probability of 0.62 or a loss of ₹ 8,000 with a probability of 0.38. Find the expected gain
What are the properties of Mathematical expectation?
Define Mathematical expectation in terms of discrete random variable
A person tosses a coin and is to receive ₹ 4 for a head and is to pay ₹ 2 for a tail. Find the expectation and variance of his gains
Choose the correct alternative:
Demand of products per day for three days are 21, 19, 22 units and their respective probabilities are 0.29, 0.40, 0.35. Profit per unit is 0.50 paisa then expected profits for three days are
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 probability density function may be represented by
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:
An expected value of a random variable is equal to it’s
Prove that V(aX) = a2V(X)
