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
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 |
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
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)
Distinguish between discrete and continuous random variables.
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
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)
The probability density function of a continuous random variable X is
f(x) = `{{:("a" + "b"x^2",", 0 ≤ x ≤ 1),(0",", "otherwise"):}`
where a and b are some constants. Find a and b if E(X) = `3/5`
The probability density function of a continuous random variable X is
f(x) = `{{:("a" + "b"x^2",", 0 ≤ x ≤ 1),(0",", "otherwise"):}`
where a and b are some constants. Find Var(X)
