math代写| Probability | 代做homework | assignment代写 – MATH-UA.0233-001 Theory of Probability

2023-05-06

MATH-UA.0233-001 Theory of Probability

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MATH-UA.0233-001 Theory of Probability homework 6

HOMEWORK assignment 6

MATH-UA.0233-001 Theory of Probability

Upload your solutions on Gradescope as a PDF file by3:00 PM on the 22nd of March. Justify all your answers.

  1. A final exam has a maximum of 30 points. Approximating the scores of the students with a normal distribution having expectation 21 and variance 9, what is the percentage of students that obtained a score 24? What is the percentage of students that obtained a score17?Approximate your final results using the table for the standard normal distribution function.
  2. LetXbe a random variable with probability density depending on the parameter > 0 given by
h(x) =
{
2 x
e

x^2
 x > 0
0 x 0

a) Verify thathis a probability density.
b) Compute expectation and variance ofX.
c) DefineZ=X^2. Compute the density ofZ, show that it follows a gamma distri-
bution (look it up on the book), and find its parameters.
  1. For >0 and >0 consider the function
f,(t) =
{
t^1 et

t > 0
0 t 0

a) Show that its a density function.
b) Compute its cumulative distribution function.
  1. LetXbe an exponential random variable with parameter >0.
a) For >0, findE[X].
b) For >0 find the density ofXand compare with exercise 4.
  1. LetXbe distributed as a standard normalN(0,1). Letg:RRbe a differentiable function and callg its derivative. Show that ifE[Xg(X)] andE[g(X)] exist, then they are equal.
  2. As practice (dont hand in any solutions for this question) read sections 5.6.3, 5.6.4, 5.6.5. Try the examples and computations on your own first, and then compare your results with those of the book.
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