Advertisements
Advertisements
प्रश्न
Suppose that the time in minutes that a person has to wait at a certain station for a train is found to be a random phenomenon with a probability function specified by the distribution function
F(x) = `{{:(0",", "for" x ≤ 0),(x/2",", "for" 0 ≤ x < 1),(1/2",", "for" ≤ x < 2),(x/4",", "for" 2 ≤ x < 4),(1",", "for" x ≥ 4):}`
What is the probability that a person will have to wait (i) more than 3 minutes, (ii) less than 3 minutes and (iii) between 1 and 3 minutes?
उत्तर
The probability that a person will have to wait
(i) more than 3 minutes
P(x > 3) = `int_2^4 "f"(x) "d"x`
= `int_3^4 (1/4) "d"x`
=`1/4(x)_3^4`
= `1/4 (4 - 3)`
= `1/4 (1)
∴ P(x > 3) = `1/4`
(ii) Less than 3 Minutes
P(x < 3) = `int_0^1 "f"(x) "d"x int_1^2 "f"(x) "d"x + int_2^3 "f"(x) "d"x`
= `int_0^1 1/2 "d"x + int_1^2 (0) "d"x + int_2^3 1/4 "d"x`
= `1/2 [x]_0^1 + 0 + 1/4 [x]2^3`
= `12 [1 - 0] + 1/4 [3 - 2]`
= `1/2 + 1/4`
=`(2 + 1)/4`
∴ P(x < 3) = `3/4`
(iii) between 1 and 3 minutes
P(1 < x < 3) = `int_1^2 "f"(x) "d"x + int_2^3 "f"(x) "d"x`
= `int_1^2 (0) "d"x + int_2^3 (1/4) "d"x`
= `0 + 1/4 (x)_2^3`
= `1/4 [3 - 2]`
= `1/4(1)`
`"P"(1 < oo < 3) = 1/4`
APPEARS IN
संबंधित प्रश्न
Construct cumulative distribution function for the given probability distribution.
| X | 0 | 1 | 2 | 3 |
| P(X = x) | 0.3 | 0. | 0.4 | 0.1 |
Explain what are the types of random variable?
Explain the terms probability distribution function
What are the properties of continuous random variable?
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:
If we have f(x) = 2x, 0 ≤ x ≤ 1, then f(x) is a
Choose the correct alternative:
The probability function of a random variable is defined as
| X = x | – 1 | – 2 | 0 | 1 | 2 |
| P(x) | k | 2k | 3k | 4k | 5k |
Then k is equal to
Choose the correct alternative:
In a discrete probability distribution, the sum of all the probabilities is always equal to
The probability distribution function of a discrete random variable X is
f(x) = `{{:(2k",", x = 1),(3k",", x = 3),(4k",", x = 5),(0",", "otherwise"):}`
where k is some constant. Find P(X > 2)
Consider a random variable X with p.d.f.
f(x) = `{(3x^2",", "if" 0 < x < 1),(0",", "otherwise"):}`
Find E(X) and V(3X – 2)
