The random variable x is exponentially distributed, where x represents the time it takes for a person to choose a birthday gift. if x has an average value of 24 minutes, what is the probability that x is less than 32 minutes? (do not round until the final step. round your answer to 3 decimal places.)
0.674
Step-by-step explanation:
If the random variable X is exponentially distributed and X has an average value of 25 minutes, then its probability density function (PDF) is
[tex]\bf f(x)=\frac{1}{25}e^{-x/25}\;(x\geq 0)[/tex]
and its cumulative distribution function (CDF) is
[tex]\bf P(X\leq t)=\int_{0}^{t} f(x)dx=1-e^{-t/25}[/tex]
So, the probability that X is less than 28 minutes is
[tex]\bf P(X\leq 28)=1-e^{-28/25}=1-e^{-1.12}=0.674[/tex]
0.685 = 68.5% probability that X is less than 30 minutes
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
If X has an average value of 26 minutes
This means that [tex]m = 26, \mu = \frac{1}{26}[/tex]
What is the probability that X is less than 30 minutes?
[tex]P(X \leq 30) = 1 - e^{-\frac{30}{26}} = 0.685[/tex]
0.685 = 68.5% probability that X is less than 30 minutes
The probability that X is less than 32 minutes is 0.736.
Step-by-step explanation:
Given : The random variable X is exponentially distributed, where X represents the time it takes for a person to choose a birthday gift. If X has an average value of 24 minutes.
To find : What is the probability that X is less than 32 minutes?
Solution :
If X has an average value of 24 minutes.
i.e. [tex]\lambda=24[/tex]
The random variable X is exponentially distributed, where X represents the time it takes for a person to choose a birthday gift.
The exponentially function is [tex]\frac{1}{\lambda}e^{-\frac{x}{\lambda}}[/tex]
The function form according to question is
[tex]f(x)=\{\frac{1}{24}e^{-\frac{x}{24}}, x0\}[/tex]
The probability that X is less than 32 minutes is
[tex]P[x[/tex]
[tex]P[x[/tex]
[tex]P[x[/tex]
Therefore, the probability that X is less than 32 minutes is 0.736.
Sorry I don’t know I’m bad at math lol