3)If the random variable T is the time to failure of a commercial product and the values of its probability density and distribution function at time "t" are f(t) and F(t), then its failure rate at time t is given by f(t) / 1-F(t)
Thus, the failure rate at time t is the probability density of failure at time t given that failure does not occur prior to time t.
a)show that if t is an exponential distribution, the failure rate is constant.
b)show that if T has a weibull distribution thae failure rate is given by (alpha*beta)*t^beta-1
a) To show that the failure rate is constant for an exponential distribution, we need to find the probability density function (PDF) and the cumulative distribution function (CDF) of the exponential distribution.
The PDF of an exponential distribution is given by f(t) = λ * e^(-λt), where λ is the rate parameter.
The CDF of an exponential distribution is given by F(t) = 1 - e^(-λt).
Now let's find the failure rate, which is given by f(t) / [1 - F(t)].
f(t) / [1 - F(t)] = (λ * e^(-λt)) / [1 - (1 - e^(-λt))]
= λ * e^(-λt) / e^(-λt)
= λ
Hence, we can see that the failure rate for an exponential distribution is constant and equal to the rate parameter λ.
b) To show that the failure rate for a Weibull distribution is given by (α * β) * t^(β-1), we need to find the PDF and CDF of the Weibull distribution.
The PDF of a Weibull distribution is given by f(t) = (β/α) * (t/α)^(β-1) * e^(- (t/α)^β), where α is the scale parameter, and β is the shape parameter.
The CDF of a Weibull distribution is given by F(t) = 1 - e^(- (t/α)^β).
Now let's find the failure rate, which is given by f(t) / [1 - F(t)].
f(t) / [1 - F(t)] = [(β/α) * (t/α)^(β-1) * e^(- (t/α)^β)] / [1 - (1 - e^(- (t/α)^β))]
= [(β/α) * (t/α)^(β-1) * e^(- (t/α)^β)] / (e^(- (t/α)^β))
= (β/α) * (t/α)^(β-1)
Hence, we can see that the failure rate for a Weibull distribution is given by (α * β) * t^(β-1), where α is the scale parameter and β is the shape parameter.