Question

Choose a numberUfrom the interval [0,1] with uniform distribution.(a) Find the cumulative distribution and density for the random variableY= ln(U+ 1).(b) What is the range of values forY?(c) Show that the density you have found has the property that∫Ωf(y)dy= 1.(d) Suppose you are handed a data sample from some statistical survey. How would youdetermine whether the sample comes from a distribution with this density?Question 4:

Answer 1

(a) Find the cumulative distribution and density for the random variable Y= ln(U+ 1).

Cumulative:

$F(y) = P(Y<y) = P(ln(U+1) < y) =P(U+1<e^{y}) = P(U<e^{y} - 1) =e^{y} - 1$

Density:

$f(y) = F'(y) = e^{y}$

(b) What is the range of values forY?

U=0 -> ln(0+1) = ln(1) = 0

U = 1 -> ln(1+1) = ln(2)

So the range of values for Y is [0, ln(2)]

(c) Show that the density you have found has the property that

∫Ωf(y)dy= 1.

$\int^{ln(2)}_{0} e^y dy = e^y |_{0}^{ln(2)} = e^{ln(2)} - e^0 = 2 - 1 = 1.$

(d) Suppose you are handed a data sample from some statistical survey. How would youdetermine whether the sample comes from a distribution with this density?

• Test hypothesis about the density of the given distribution
• Plus Hypothesis about maximum and minimum value of the Uniform distribution:

$X_{max} =\underset{i} {max}( e^{X_i} - 1)$

$X_{min} =\underset{i} {min}( e^{X_i} - 1)$