Advertisements
Advertisements
प्रश्न
The distribution of a continuous random variable X in range (– 3, 3) is given by p.d.f.
f(x) = `{{:(1/16(3 + x)^2",", - 3 ≤ x ≤ - 1),(1/16(6 - 2x^2)",", - 1 ≤ x ≤ 1),(1/16(3 - x)^2",", 1 ≤ x ≤ 3):}`
Verify that the area under the curve is unity.
उत्तर
Since(– 3, 3) in the range of given p.d.f
Area under the curve A = `int_oo^oo "f"(x) "d"x = int_3^3 "f"(x) "d"x`
A = `int_(-3)^(-1) "f"(x) "d"x + int_(-1)^1 "f"(x) "d"x + int_1^3 "f"(x) "d"x`
A = `int_(-3)^(-1) 1/16 (3 + x)^2 "d"x + int_(-1)^1 1/16 (6 - 2x^2) "d"x + int_1^3 1/16 (3 - x)^2 "d"x`
= `1/16{[(3 + x)^3/3]_(-3)^(-1) + [6x - (3x^3)/3]_(-1)^1 + [(3 - x)^3/(3(-1))]_1^3}`
= `1/16 {(3 - 1)^3/3 - (3 - 3)^3/3 + [6 - 2/3] - [-6 + 2/3] + [((3 - 3)/(-3))^3] - ((3 - 1)/(-3))^3}`
= `1/16 {(2)^3/3 - 0/3 + 6 - 2/3 + 6 - 2/3 + 0 + (2)^3/3}`
= `1/16{8/3 + 6 - 2/3 + 6 - 2/3 + 8/3}`
= `1/16 {12 + 12/3}`
= `1/16 {12 + 4}`
= `1/16 {16}`
= 1
∴ A = 1 sq.units
Hence the Area under the curve is unity.
APPEARS IN
संबंधित प्रश्न
In a pack of 52 playing cards, two cards are drawn at random simultaneously. If the number of black cards drawn is a random variable, find the values of the random variable and number of points in its inverse images
Two balls are chosen randomly from an urn containing 6 red and 8 black balls. Suppose that we win ₹ 15 for each red ball selected and we lose ₹ 10 for each black ball selected. X denotes the winning amount, then find the values of X and number of points in its inverse images
Choose the correct alternative:
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. Then the possible values of X are
The discrete random variable X has the probability function
| X | 1 | 2 | 3 | 4 |
| P(X = x) | k | 2k | 3k | 4k |
Show that k = 0 1
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 |
Find k
Explain the terms probability distribution function
Choose the correct alternative:
A variable that can assume any possible value between two points is called
Choose the correct alternative:
A formula or equation used to represent the probability distribution of a continuous random variable is called
Choose the correct alternative:
The probability density function p(x) cannot exceed
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(X ≤ 0)
