Answer to Question #235300 in Statistics and Probability for Hastings Mambwe

Question

b) The life of an electronic device is known to have the exponential distribution

with parameter  = 1

1000 .

(i) What is the probability that the device lasts more than 1000 hours?

[2 marks]

(ii) What is the probability it will last less than 1200 hours?

[2 marks]

(iii) Find the mean and variance of the life of the electronic device.

[4 marks]

Answer 1

Let $X=$ the life of an electronic device: $X\sim Po(\lambda).$

Given $\lambda=1000.$

For sufficiently large values of $λ,$ (say $λ>1,000$ ), the $Normal(μ = λ,σ^2 = λ)$ Distribution is an excellent approximation to the Poisson$(λ)$ Distribution.

(i)

$$P(X>1000)=1-P(X\leq 100)$$

$$\approx1-P(Z\leq\dfrac{1000+0.5-1000}{\sqrt{1000}})$$

$$\approx1-P(Z\leq 0.01581139)\approx0.49369243$$

(ii)

$$P(X<1200)=P(Z<\dfrac{1200-0.5-1000}{\sqrt{1000}})$$

$$\approx P(Z<6.30874393)$$

$$\approx0.99999999986\approx1$$

(iii)

$$\mu=\lambda=1000$$

$$Var(X)=\sigma^2=\lambda=1000$$