Question

A lamp has two bulbs, each of a type with an average lifetime of 9 hours. The probability density function for the lifetime of a bulb is f(t)=19ᵉ−ᵗ/9,t≥0. What is the probability that both of the bulbs will fail within 5 hours?

Answer 1

The probability that both of the bulbs will fail within 5 hours is 0.287.

How to find probability?

To find the probability that both bulbs will fail within 5 hours, find the joint probability density function and then integrate it over the given range.

The probability density function (pdf) for the lifetime of a single bulb is given by:

`\[ f(t) = (1)/(9) e^(-t/9), \quad t \geq 0 \]`

To find the joint probability density function for both bulbs failing within a certain time, multiply the individual pdfs. Let T₁ be the lifetime of the first bulb, and T₂ be the lifetime of the second bulb. The joint pdf is given by:

`\[ f_(T_1, T_2)(t_1, t_2) = f(t_1) \cdot f(t_2) \]`

Now, substitute the given pdf into this expression:

`\[ f_(T_1, T_2)(t_1, t_2) = (1)/(81) e^(-t_1/9) e^(-t_2/9) \]`

The integral for the joint probability is:

`\[ P(T_1 \leq 5, T_2 \leq 5) = \int_(0)^(5) \int_(0)^(5) (1)/(81) e^(-t_1/9) e^(-t_2/9) \, dt_1 \, dt_2 \]`

Calculate it:

`\[ P(T_1 \leq 5, T_2 \leq 5) = \int_(0)^(5) \int_(0)^(5) (1)/(81) e^(-t_1/9) e^(-t_2/9) \, dt_1 \, dt_2 \]`

`\[ = (1)/(81) \int_(0)^(5) \int_(0)^(5) e^(-t_1/9) e^(-t_2/9) \, dt_1 \, dt_2 \]`

`\[ = (1)/(81) \int_(0)^(5) \left( \int_(0)^(5) e^(-t_1/9) e^(-t_2/9) \, dt_2 \right) dt_1 \]`

`\[ = (1)/(81) \int_(0)^(5) \left( \int_(0)^(5) e^(-(t_1 + t_2)/9) \, dt_2 \right) dt_1 \]`

`\[ = (1)/(81) \int_(0)^(5) \left( -9e^(-(t_1 + t_2)/9) \Big|_0^5 \right) dt_1 \]`

`\[ = (1)/(81) \int_(0)^(5) \left( -9e^(-5/9) + 9 \right) dt_1 \]`

`\[ = (1)/(81) \left( -9e^(-5/9) + 9 \right) \int_(0)^(5) dt_1 \]`

`\[ = (1)/(81) \left( -9e^(-5/9) + 9 \right) t_1 \Big|_0^5 \]`

`\[ = (1)/(81) \left( -9e^(-5/9) + 9 \right) \cdot 5 \]`

`\[ = (1)/(81) \left( -45e^(-5/9) + 45 \right) \]`

The calculated value for
`\( P(T_1 \leq 5, T_2 \leq 5) \)` is approximately 0.287.