Advertisements
Advertisements
प्रश्न
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"):}`
What is the probability that the number of minutes that person will talk over the phone is (i) more than 10 minutes, (ii) less than 5 minutes and (iii) between 5 and 10 minutes
Advertisements
उत्तर
(i) more than 10 minutes
`int_10^00 "f"(x) "d"x`
= `1/5 int_10^oo "e"^(x/5) "d"x`
= `1/5 ("e"^((-x)/5)/(((-1)/5)))^oo`
= `- ["e"^((-x)/5)]_10^oo`
= `- ["e"^-oo - "e"^((-10)/5)]`
= `- [0 - "e"^-2]`
= `"e"^-2`
= `1/"e"^2`
(ii) less than 5 minutes
`int_0^5 f(x) "d"x = int_0^5 "Ae"^((x)/5)`
= `1/5 int_0^5 "e"^((-x)/5) "d"x`
= `1/5 ["e"^((-x)/5)/((-1)/5)]_0^5`
= `- ["e"^((-x)/5)]_0^5`
= `- ["e"^((-5)/5) - "e"^0]`
= `- ("e"^-1 - 1)`
= `1 - "e"^-1`
= `1 - 1/"e"`
= `("e" - 1)/"e"`
(iii) between 5 and 10 minutes
`int__5^10 "f"(x) "d"x = int_5^10 "Ae"^((-x)/5) "d"x`
= `int_5^10 1/5 "e"^((-x)/5) "d"x`
= `1/5 ["e"^((-x)/5)/((-1)/5)]_5^10`
= `- ["e"^((-x)/5)]_5^10`
= `- ["e"^((-10)/5) - "e"^((-5)/5)]`
= `[-"e"^-2 - "e"^-1]`
= `"e"^-1 - "e"^-2`
= `1/"e"- 1/"e"^2`
= `("e" - 1)/"e"^2`
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 |
Let X be a discrete random variable with the following p.m.f
`"P"(x) = {{:(0.3, "for" x = 3),(0.2, "for" x = 5),(0.3, "for" x = 8),(0.2, "for" x = 10),(0, "otherwise"):}`
Find and plot the c.d.f. of X.
Two coins are tossed simultaneously. Getting a head is termed a success. Find the probability distribution of the number of successes
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):}`
Is the distribution function continuous? If so, give its probability density function?
Explain what are the types of random variable?
Explain the terms probability distribution function
Choose the correct alternative:
If c is a constant in a continuous probability distribution, then p(x = c) is always equal to
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| ≤ 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)
