Question

The lifetime in months of a transistor in a certain application is random with probability density function:

Answer 1

Answer:

Explained below.

Step-by-step explanation:

The probability density function of lifetime in months of a transistor in a certain application is:

$f(x)=0.4\cdot e^{-0.4x};x>0$

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:

$\mu=\frac{1}{\lambda}=\frac{1}{0.40}=2.5$

Thus, the mean lifetime is 2.5 months.

(b)

Compute the standard deviation of the lifetimes as follows:

$\sigma=\sqrt{\frac{1}{\lambda^{2}}}=\sqrt{\frac{1}{0.40^{2}}}=2.5$

Thus, the standard deviation of the lifetimes is 2.5 months.

(c)

The cumulative distribution function of the lifetime is:

$CDF=1-e^{-0.40x}$

(d)

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

$P(X$

               $=0.40\times [\frac{e^{-0.40x}}{-0.40}]^{6}_{0}\\\\=-1\times [e^{-0.40\times6}-e^{-0.40\times0}]\\\\=-0.090718+1\\\\=0.909282\\\\\approx 0.9093$

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:

$P(X$

Thus, the 60th percentile of the lifetime is 2.3 months.