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