Consider a function f(x) = 0.4 x € (0,1) 0.2 x = (1,4) 0x (0,4). This function gives rise to a Probability Density Function Compute P(1.30 < x < 3.58)
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- Recall that the general form of a logistic equation for a population is given by P(t)=c1+aebt , such that the initial population at time t=0 is P(0)=P0. Show algebraically that cP(t)P(t)=cP0P0ebt .Repeat Example 5 when microphone A receives the sound 4 seconds before microphone B.QUESTION 11 The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a ≤ x ≤ b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices.
- Question 10 The joint probability density function of X and Y is given byQuestion 1.2 Consider the function f (x) = (1/24(x^2 +1) 1 < or = x < or = 4) = (0 otherwise) Calculate P (x = 3) Calculate P (2 < or = x < or = 3) Question 1.3 Consider the function f (x) = (k - x/4 1 < or = x < or = 3) = (0 otherwise) which is being used as a probability density function for a continuous random variable x? a. Find the value of K b. Find P (x < or = 2.5)Theorem A at Section 8.8.1 saysA necessary and sufficient condition for T(X1,..., Xn) to be sufficient for a parameter θ is that the joint probability function (density function or frequency function) factors in the formf(x1,...,xn|θ) = g[T(x1,...,xn),θ]h(x1,...,xn)
- QUESTION 10 Suppose f(x) = 1/4 over the range a ≤ x ≤ b, and suppose P(X > 4) = 1/2. What are the values for a and b? a. 2 and 6 b. Cannot answer with the information given. c. 0 and 4 d. Can be any range of x values whose length (b − a) equals 4. QUESTION 11 The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a ≤ x ≤ b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices. QUESTION 12 Which of the following is true about f(x) when X has a uniform distribution over the interval [a, b]? a. The values of f(x) are different for various values of the random variable X. b. f(x) equals one for each possible value of X. c. f(x) equals one divided by the length of the interval from a to b.…Suppose ? and ? have joint probability density function ?(?, ?) = { ?(?^2 + ???), 0 ≤ x ≤ 1, −1 ≤ ? ≤ 1, 1 ≤ ? ≤ 2 0, elsewhere. a. Find ?. b. Find ?(0.5 < ?, 0 < ? ≤ 1, ? < 1.5. Please answer this question step by step and justify your solutionsQuestion 1 : Suppose that the probability density function (p.d.f.) of the life (in weeks) of a certain part is f(x) = 3 x 2 (400)3 , 0 ≤ x < 400. (a) Compute the probability the a certain part will fail in less than 200 weeks. (b) Compute the mean lifetime of a part and the standard deviation of the lifetime of a part. (c) To decrease the probability in part (a), four independent parts are placed in parallel. So all must fail, if the system fails. Let Y = max{X1, X2, X3, X4} denote the lifetime of such a system, where Xi denotes the lifetime of the ith component. Show that fY (y) = 12 y 11 (400)12 , y > 0. Hint : First construct FY (y) = P(Y ≤ y), by noticing that {Y ≤ y} = {X1 ≤ y} ∩ {X2 ≤ y} ∩ {X3 ≤ y} ∩ {X4 ≤ y}. (d) Determine P(Y ≤ 200) and compare it to the answer in part (a)
- What is the value if the constant c for p(x) to qualify as a probability mass function? p(x)=c(1/4)x-1 if x = 1, 2, 3, 4, 5, ... and p(x) = 0 otherwise.1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.Consider the following constant function: f(x) = 3/4 If this function represents a probability density function, which of the following can be a set of realized values? A. 2/3 ≤ x ≤ 2 B. 1/4 ≤ x ≤ 1 C. 3 ≤ x ≤ 4 D. 1/2 ≤ x ≤ 3/2
