Answer:
1. 0.0498
2. 0.8775
Explanation:
Let X represent the time of failures (in hours) of PC fans
From the question, X is exponentially distributed with the parameter λ = 0.0003
Also, it's to be noted that X is a continuous random variable
The probability distribution function of X is f(x) = λe^-(λx) for x >= 0
To solve this, we need to get the CDF (cumulative distribution function) of the above PDF
CDF
The CDF is simply the integration (integral calculus) of the PDF
Let F(x) represent the CDF
1. What proportion of the fans will last at least 10,000 hours
F(x) = 1 - e^-(λx) for x>=0
For x = at least 10000 hours and λ = 0.0003 -------- The keyword here is at least
Because P(True) + P(False) = 1 ----------- (Probability of true + Probability of false = 1),
We have
F(x>= 10000) = 1 - F(x<= 10000) -------- Notice the change in the inequality sign
This then equates to
F(x>=10000) = 1 - (1 - e^-(λx)) ---------- Open the bracket
F(x>=10000) = 1 - 1 + e^-(λx)
F(x>=10000) = e^-(λx)
F(x>=10000) = e^-(0.0003 * 10000)
F(x>=10000) = e^-(3)
F(x>=10000) = 0.049787068367863
F(x>=10000) = 0.0498 ---------- Approximated
2. What is the proportion of the fans will last at most 7000 hours
X = at most 7000 and λ = 0.0003
The keyword here is at most, so we make use of
F(x) = 1 - e^-(λx) for x>=0
F(x<=7000) = 1 - e^-(λx)
F(x<=7000) = 1 - e^-(0.0003 * 7000)
F(x<=7000) = 1 - e^-(2.1)
F(x<=7000) = 1 - 0.122456428252981
F(x<=7000) = 0.877543571747018
F(x<=7000) = 0.8775 ---------- Approximated
