Probstat/week8 practice 1

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This exercise considers continuous random variables.

Probability

1. Let random variable is a continuous random variable which is a real number chosen uniformly from the range (10,50). Let denote its probability distribution function.

  • 1.1 Describe .
  • 1.2 What is .
  • 1.3 What is . (Note that this is a conditional probability.)

2. Let random variable is a continuous random variable which is a real number chosen uniformly from the range (a,b). Let denote its probability distribution function.

  • 2.1 Describe .
  • 2.2 Find the value of such that .

Definition: A cumulative distribution function (or cdf in short) for a continuous random variable whose probability distribution function is is defined to be

3. Let random variable is a continuous random variable whose probability distribution function is given by

  • 3.1 Find the value of c.
  • 3.2 Plot pdf of X.
  • 3.3 Let be 's cumulative distribution function. Find .
  • 3.4 Plot

4. (FCP. Example 1a.) Suppose that X is a continuous random variable whose pdf is given by

  • 4.1 Find .
  • 4.2 Find .

5. (FCP. Example 1b.) The time (in hours) that a computer functions before breaking down is a continuous random variable with probability density function given by

  • 5.1 Find the value of
  • 5.2 What is the probability that a computer will function between 50 and 100 hours before breaking down?
  • 5.3 What is the probability that it will function less than 100 hours?

Hint: Note that

6. A random variable has pdf defined by

  • 6.1 Find .
  • 6.2 Find its cdf and plot it.
  • 6.3 What is ?

Expectation

Definition: If is a continuous random variable whose probability distribution function (pdf) is , we define the expected value of to be

1. Find for continuous random variable defined in question 1 in the previous section.

2. Find for continuous random variable defined in question 2 in the previous section.

3. Find for continuous random variable defined in question 3 in the previous section.

4. Find for continuous random variable defined in question 4 in the previous section.

5. Find for continuous random variable defined in question 5 in the previous section.

Hint: Note that

6. Find for continuous random variable defined in question 6 in the previous section.

7. Find and Var() for continuous random variable defined in question 1 in the previous section.

8. Find and Var() for continuous random variable defined in question 2 in the previous section.

9. Find and Var() for continuous random variable defined in question 3 in the previous section.