Answer:
Explained below.
Explanation:
The probability density function of lifetime in months of a transistor in a certain application is:

The probability density function suggests that the random variable X follows a exponential function with parameter λ = 0.4.
(a)
Compute the mean lifetime as follows:

Thus, the mean lifetime is 2.5 months.
(b)
Compute the standard deviation of the lifetimes as follows:

Thus, the standard deviation of the lifetimes is 2.5 months.
(c)
The cumulative distribution function of the lifetime is:

(d)
Compute the probability that the lifetime will be less than 6 months as follows:

![=0.40* [(e^(-0.40x))/(-0.40)]^(6)_(0)\\\\=-1* [e^(-0.40*6)-e^(-0.40*0)]\\\\=-0.090718+1\\\\=0.909282\\\\\approx 0.9093](https://img.qammunity.org/2021/formulas/mathematics/high-school/js128cmpuauu4kbc0c4m6t7m8hdwowat8e.png)
Thus, the probability that the lifetime will be less than 6 months is 0.9093.
(e)
Compute the 60th percentile of the lifetime as follows:
![P(X<x)=0.60\\\\\int\limits^(x)_(0) {0.40\cdot e^(-0.40x)} \, dx=0.60\\\\0.40* [(e^(-0.40x))/(-0.40)]^(x)_(0)=0.60\\\\-1* [e^(-0.40* x)-e^(-0.40*0)]=0.60\\\\1-e^(-0.40x)=0.60\\\\e^(-0.40x)=0.40\\\\-0.40x=\ln(0.40)\\\\-0.40x=-0.916291\\\\x=2.2907275\\\\x\approx 2.3](https://img.qammunity.org/2021/formulas/mathematics/high-school/a971avg0vsdp1me4jso45pz88ho5bzqrsc.png)
Thus, the 60th percentile of the lifetime is 2.3 months.