STAT 5100 Midterm: STAT 5100 Iowa Midterm2

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turquoisegnu128

31 Jan 2019

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Please write your answers in the exam books provided. You can use formulas from page 2 when appropriate: (5 points) let x be a continuous random variable with density fx(x) =( 3. Compute the mean and variance of x: (5 points) let x and y be jointly continuous with joint density f (x, y)(c(x + y)e (x+y) 0 for x 0, y 0 otherwise for some c > 0. Find the value of c and the marginal density of u = (x +y )/2. Find the correlation of x and y . [hint: use properties of conditional expectations, such as the law of total expectation and the variance decomposition formula, for your calculations. Compute the moment generating function of the marginal distribution of x and use it to identify this marginal distribution. P (x = x|p) = px(1 p)1 x; x = 0, 1; 0 p 1 x(cid:1)px(1 p)n x; x = 0, 1, .

Related Questions

Q5. Let (Ω, F , P) be a probability space. Let λ > 0 be some constant.

(a) Give the definition of a random variable X on R ∪ {−∞, +∞}.
(b) Define the distribution function FX for the RV X.

[6 marks]

[1 marks]

[1 marks] (c) Suppose that Y is a RV on R ∪ {−∞, +∞} with distribution function given by

􏰄3 −1e−λy (y≥0)

[6 marks]

FY (y) = 4 2 1

(y < 0).
For any a < b, determine P(Y > a), and P(Y ∈ (a, b]). [3 marks]

(d) By noting {Y < b} = ∪n∈N{Y ≤ b − 1 } and using Continuity of Probability, n

find P(Y < b). Similarly, find P(Y < ∞) and P(Y = −∞). [4 marks]
(e) Is Y either a discrete RV, or a continuous RV, or a mixture of both? Explain

your answer. [1 marks]

amaranthwalrus338

1. In its standardized form, the normal distribution

(a) has a mean of 0 and a standard deviation of 1.

(b) has a mean of 1 and a variance of 0.

(d) cannot be used to approximate discrete probability distributions.

2. For some positive value of Z, the probability that a standard normal variable is between 0 and Z is 0.3770.

The value of Z is

(a) 0.18

(b) 0.81

(d) 1.47

3. For some value of z, the probability that a standard normal variable is below z is 0.2090. The value of Z is

(a) -0.81

(b) -0.31

(d) 1.96

4. For some positive value of X, the probability that a standard normal variable is between 0 and +2x is 0.1255.

The value of X is

(a) 0.99

(b) 0.40

(d) 0.16

5. If we know that the length of time it takes a college student to find a parking spot in the library parking lot follows a normal distribution with a mean of 35. minutes and a standard deviation of minute, find the probability that a randomly selected college student will find a parking spot in the library parking lot in less than 3 minutes.

(a) 0.3551

(b) 0.3085

(d) 0.1915