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W06 - HW Solutions

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Normal distribution

01 - Converting to standard normal

Let . Show that is a standard normal random variable.

In other words, verify that the PDF of is .

Hint: Use a method analogous to one in the lecture notes.

Solution

  1. & Express in terms of .
  2. & Compute .
  3. & State formula for finding the PDF of .
  4. & State PDF for .
  5. & Compute PDF for using the formula.

02 - Symmetry of

Show that for all .

Solution

  1. & Recall the definition of .
  2. Use -substitution to simplify the integral.
    • Let
    • Change the bounds, and .
  3. & Recall that integrating a PDF from to yields .

03 - Generalized normal - misc

Let be generalized normal variable with and . Using a chart of values, find:

  • (a)
  • (b) such that
  • (c) (Hint: Use and to avoid integration.)

Solution (a)

  1. & Write desired probability in terms of values.
  2. & Evaluate using a value table.

(b)

  1. & Use the value table to find if
  2. & Solve for knowing that .

(c)

  1. & Recall formula for and solve for
  2. Plug in for and for and compute .

04 - Normal distribution - test scores

In a large probability theory exam, the scores are normally distributed with a mean of 75 and a standard deviation of 10.

  • (a) What is the probability that a student scored between 70 and 80?
  • (b) What is the lowest score a student can achieve to be in the top 5?
  • (c) What score corresponds to the 25th percentile?

Solution (a)

  1. & Write desired probability in terms of values.
  2. & Use table to evaluate expression.

(b)

  1. & Interpret problem.
    • Since we wish to find the top , we wish to find such that .
  2. & Use lookup table to find .
  3. & Given that , solve for .

(c)

  1. & Interpret problem.
    • Since we wish to find the 25th percentile, we wish to find such that .
  2. & Use lookup table to find .
  3. & Given that , solve for .

05 - Normal distribution - cars passing toll booth

The number of cars passing a toll booth on Wednesdays has a normal distribution .

  • (a) What is the probability that on a randomly chosen Wednesday, more than 1,400 cars pass the toll booth?
  • (b) What is the probability that between 1,000 and 1,400 cars pass the toll booth on a random Wednesday?
  • (c) Suppose it is also known that at least 1200 cars passed the toll booth last Wednesday. What is the probability that at least 1300 cars passed the toll booth that day?

Solution (a)

  1. Write desired probability in terms of values.
  2. Use lookup table to compute probability.

(b)

  1. Write desired probability in terms of values.
  2. Use lookup table to compute probability.

(c)

  1. Set up conditional probability expression.
  2. Write desired probability in terms of values.
  3. Use lookup table to compute probability.