Let X denote an exponential random variable with unknown parameter λ>0 . Let Y=I(X>5) , the indicator that X is larger than 5 .
Recall the definition of the indicator function here is
I(X>5)={1ifX>50ifX≤5.
We think of Y as a censored version of the Exponential random variable X : we cannot directly observe X , but we are able to gather some information about it (in this case, whether or not X is larger than 5 .)
Observe that Y is a Bernoulli random variable. Thus, the statistical model for Y can be written ({0,1},{Ber(f(λ))}λ>0) for some function f of λ . What is f(λ) ?
To find the function f(λ) that represents the statistical model for Y, we need to determine the probability that Y takes on the value 1 (indicating that X is larger than 5) for a given value of λ.
Since Y is defined as the indicator function I(X>5), we need to find P(X>5) for a given value of λ.
To do this, we can use the cumulative distribution function (CDF) of the exponential distribution. The CDF of an exponential distribution with parameter λ is given by:
F(x) = 1 - e^(-λx)
To find P(X>5), we subtract the CDF at 5 from 1:
P(X>5) = 1 - F(5) = 1 - (1 - e^(-5λ)) = e^(-5λ)
Therefore, the function f(λ) that represents the statistical model for Y is:
f(λ) = e^(-5λ)
The function f(λ) represents the probability of success in a Bernoulli random variable, given the parameter λ. In this case, the probability of success is the probability that X is larger than 5, which can be denoted as P(X > 5).
Recall that for an exponential random variable with parameter λ, the probability density function (pdf) is given by f(x) = λ * e^(-λx) for x ≥ 0.
To find the probability that X is larger than 5, we need to calculate the integral of the pdf from 5 to infinity, and subtract it from 1:
P(X > 5) = 1 - ∫[5 to ∞] λ * e^(-λx) dx
Let's calculate this integral step-by-step.
Step 1: Rewrite the integral:
P(X > 5) = 1 - ∫[5 to ∞] λ * e^(-λx) dx
= 1 - [-e^(-λx)] [5 to ∞]
Step 2: Evaluate the integral limits:
P(X > 5) = 1 - [-e^(-λ∞)] + e^(-λ*5)
Step 3: Simplify the expression:
P(X > 5) = 1 - 0 + e^(-λ*5)
= 1 + e^(-λ*5)
Therefore, the function f(λ) for the Bernoulli random variable Y is f(λ) = 1 + e^(-λ*5).