Question 3: In this exercise you will use R to compute the normal density and the cumulative distribution functions using dnorm and pnorm. (a) For any given values of μ, σ, and y, what value does dnorm(y, μ,o) compute? Write the expression in terms of μ, and y. How does the value compare to the value computed by 1/0+dnorm((y-μ)/o)? Explain. (b) For any given values of μ, σ, and y, what value does pnorm(y,μ, o) compute? Write the expression in terms of μ, and y.How does the value compare to the value computed by pnorm((y-μ)/o)? Explain.
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- For a laboratory assignment, if the equipment is working, the density function of the observed outcome X is f(x) = 2x(1-x), 0 < x < 1 Find the variance and standard deviation of XFor a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)The Weibull distribution has the following density functionf(x) = (α / βα)*xα-1*e-(x/β)^α, y ≥ 0 (Shown in picture) 1. Find the cumulative distribution function and express the median as a function of the parameters α and β.2. Under what condition will this density function equal the exponential density function?3. What would be the expression for the joint distribution of 10 independent random variables with the Weibull density function? Summarize it as much as possible.
- Suppose the random variable Y has an exponential distribution with mean 2. Use the method of transformations to find the density function of U=(1/5)Y+3The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following density function. What is P(X ≤ 4)? What is P(4.5 < X)?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.
- Example 2: For each part below, find the endpoints on a standard normal density (curves have X~N(0,1) ). The area to the left of the endpoint is about 0.20. The area between ± z is 0.80. The area to the right of the endpoint is 0.4A continuous variable Y has a probability density function for which the moment generating function is given by M(t)=e^(2*t+72*t^2). What is the variance of the variable, Var[Y]?Find the mean, standard deviation and the median of the density function: f(x)= (6/343)x(7-x) [0,7]
- Assume that the range of X is [4.9, 5.11 mA, and assume that the probability density function of X is f(x) = 5 for 4.9 <= x <= 5.1. What is the variance?Suppose that Y1, . . . , Yn is a random sample from a population whose density function is1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.