Suppose that T (≥ 0) is a continuous random variable. Let its pdf, cdf, survival function,...

Suppose that T (≥ 0) is a continuous random variable. Let its pdf, cdf, survival function, hazard rate function, and cumulative hazard rate function be f(t), F(t), s(t), h(t), and H(t), respectively. Note that H(t) is defined by R t 0 h(u)du.

a. Denote s(t), h(t), and H(t) as a function of f(t).

b. Denote f(t), h(t), and H(t) as a function of s(t).

c. Denote f(t), s(t), and H(t) as a function of h(t).

d. Denote f(t) and s(t) as a function of H(t).

Expert Solution

milcah answered 1 year ago

Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that...

Let T > 0 be a continuous random variable with cumulative hazard function H(·). Show that H(T)∼Exp(1),where Exp(λ) in general denotes the exponential distribution with rate parameter λ. Hints: (a) You can use the following fact without proof: For any continuous random variable T with cumulative distribution function F(·), F(T) ∼ Unif(0,1). Hence S(T) ∼ Unif(0,1). (b) Use the relationship between S(t) and H(t) to derive that Pr(H(T) > x)= e−x.

(15 pts) Suppose that the continuous random variable X has pdf ?(?) = { ?; 0...

(15 pts) Suppose that the continuous random variable X has pdf ?(?) = { ?; 0 < ? < 2 2?; 5 < ? < 10 0; otherwise a) Determine the value of c that makes this a legitimate pdf. b) Sketch a graph of this pdf. c) Determine the cumulative distribution function (cdf) of X. d) Sketch a graph of this cdf. e) Calculate ? = ?(?) and ? = ??(?). f) What is ?(? = ?)? g) Compute...

Let X be a continuous random variable with pdf: f(x) = ax^2 − 2ax, 0 ≤...

Let X be a continuous random variable with pdf: f(x) = ax^2 − 2ax, 0 ≤ x ≤ 2 (a) What should a be in order for this to be a legitimate p.d.f? (b) What is the distribution function (c.d.f.) for X? (c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)? (d) What is the 90th percentile value of this distribution? (Note: If you do this problem correctly, you will end up with a cubic...

2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 − x), 0 <...

2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 − x), 0 < x < 1, 0 elsewhere. (a) Find the value of c such that fX(x) is indeed a PDF. (b) Find P(−0.5 < X < 0.3). (c) Find the median of X.

Suppose that a continuous, positive random variable T has a hazard function defined by h(t) =...

Suppose that a continuous, positive random variable T has a hazard function defined by h(t) = hj when aj-1 ≤ t < aj, j = 1, 2, …, k, where a0 = 0 and ak= ∞, a0 < a1 < … < ak and hj> 0, j = 1, 2, …, k. (a) Express S(t), the survivor function of T, in terms of the hj’s. (b) Express f(t), the probability density function of T, in terms of the hj’s. (c)...

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0,...

Let X be a exponential random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x). (a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks] (b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships,...

Let X be a uniform random variable with pdf f(x) = λe−λx for x > 0,...

Let X be a uniform random variable with pdf f(x) = λe−λx for x > 0, and cumulative distribution function F(x). (a) Show that F(x) = 1−e −λx for x > 0, and show that this function satisfies the requirements of a cdf (state what these are, and show that they are met). [4 marks] (b) Draw f(x) and F(x) in separate graphs. Define, and identify F(x) in the graph of f(x), and vice versa. [Hint: write the mathematical relationships,...

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0...

Let X and Y be continuous random variables with joint pdf f(x, y) = kxy^2 0 < x, 0 < y, x + y < 2 and 0 otherwise 1) Find P[X ≥ 1|Y ≤ 1.5] 2) Find P[X ≥ 0.5|Y ≤ 1]

Let X be a random variable. Suppose X ∼ Exp(0.5). The area under the pdf of...

Let X be a random variable. Suppose X ∼ Exp(0.5). The area under the pdf of the Exp(0.5) distribution above the interval [0, 2] is 0.6321. What is the value of the cumulative distribution function FX of X at x = 2?

Let X be a continuous random variable that has a uniform distribution between 0 and 2...

Let X be a continuous random variable that has a uniform distribution between 0 and 2 and let the cumulative distribution function F(x) = 0.5x if x is between 0 and 2 and let F(x) = 0 if x is not between 0 and 2. Compute 1. the probability that X is between 1.4 and 1.8 2. the probability that X is less than 1.2 3. the probability that X is more than 0.8 4. the expected value of X...

Question