Advertisements
Advertisements
Question
The probability density function of a continuous random variable X is
f(x) = `{{:(a + bx^2",", 0 ≤ x ≤ 1),(0",", "otherwise"):}`
where a and b are some constants. Find Var(X)
Advertisements
Solution
`"E"(x^2) = int_(-oo)^oo x^2"f"(x) "d"x`
= `int_0^1 ("a"x^2 + "b"x4) "d"x`
= `{3/5 (x^3/3) + 6/5[x^5/5]}_0^1`
= `[1/5 (x^3) + 6/25 (x^5)]_0^1`
= `[1/5 (1) + 6/25(1)] - [0]`
= `1/5 + 6/25`
= `(5 + 6)/25`
= 11
Var(x)= `"E"(x^2) - ["E"(x)]`
= `1/25 - (3/5)^2`
= `11/25 - 9/5`
= `(11 - 9)/25`
Var(x) = `2/25`
APPEARS IN
RELATED QUESTIONS
An urn contains 5 mangoes and 4 apples. Three fruits are taken at random. If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images
Construct cumulative distribution function for the given probability distribution.
| X | 0 | 1 | 2 | 3 |
| P(X = x) | 0.3 | 0. | 0.4 | 0.1 |
The discrete random variable X has the following probability function.
P(X = x) = `{{:("k"x, x = 2"," 4"," 6),("k"(x - 2), x = 8),(0, "otherwise"):}`
where k is a constant. Show that k = `1/18`
The discrete random variable X has the probability function.
| Value of X = x |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| P(x) | 0 | k | 2k | 2k | 3k | k2 | 2k2 | 7k2 + k |
Evaluate p(x < 6), p(x ≥ 6) and p(0 < x < 5)
The length of time (in minutes) that a certain person speaks on the telephone is found to be random phenomenon, with a probability function specified by the probability density function f(x) as
f(x) = `{{:("Ae"^((-x)/5)",", "for" x ≥ 0),(0",", "otherwise"):}`
Find the value of A that makes f(x) a p.d.f.
Explain the distribution function of a random variable
Choose the correct alternative:
If c is a constant, then E(c) is
Choose the correct alternative:
A set of numerical values assigned to a sample space is called
The probability function of a random variable X is given by
p(x) = `{{:(1/4",", "for" x = - 2),(1/4",", "for" x = 0),(1/2",", "for" x = 10),(0",", "elsewhere"):}`
Evaluate the following probabilities
P(0 ≤ X ≤ 10)
The p.d.f. of X is defined as
f(x) = `{{:("k"",", "for" 0 < x ≤ 4),(0",", "otherwise"):}`
Find the value of k and also find P(2 ≤ X ≤ 4)
