Homework 11

Problems

Additional Instruction: For each of the following problems that ask for the probability that a continuous random variable falls in some interval:

1. sketch the probability density function for the random variable;
2. shade the query region under consideration on the horizontal axis;
3. shade the area under the probability density function that corresponds to the probability that the random variable falls in the query region.

You may use any graphing utility (Excel, graphing calculator, R, WolframAlpha, etc.) to get an idea of the shape of the probability density function. I recommend Desmos. You can find a demonstration of using Desmos with Problem 4.1.4 here.

Section 4.1:

• 4; Hint: use \(u\)-substitution to find the antiderivative.
• 6
• 12(a-d)

Section 4.2:

• 22(c)
• 25; Hint: how is the area \(A\) of the circle related to \(R\)?

Additional Problem:

Show that, for a continuous random variable \(X\),

• \(E[a X + b] = a E[X] + b\)
• \(\text{Var}(a X + b) = a^{2} \text{Var}(X)\)