Cumulative Distribution Functions

We've talked about the probability density function of a random variable. There is another function associated with a random variable that is often useful as well: the cumulative distribution function (cdf). The cdf FX of the random variable X is defined as F X (x) = P(X ≤ x) Exercises: 1. a. The diagram shows the graph of the pdf f X (x) of the continuous random variable X. How can you draw something in the picture that shows F X (c), the value of the cdf of X at c? [Hint: Remember the definition of the pdf.] c b. Use the idea in part (a) to give a formula for finding F X (c) in terms of f X (x) (still assuming X is a continuous random variable). 2. If X is a continuous random variable and you know its cdf, how can you find its pdf? [Hint: Use Exercise 1(b).] 3. If a < b, what can you say about the relationship between F X (a) and F X (b)? (Your answer and reasoning should just depend on the definition of cdf, not on whether X is discrete or continuous.) 4. X is a discrete random variable that only takes on values 0, 1, 2, and 4, with probabilities ½, ¼, 1/8, and 1/8, respectively. What is the cdf of X? Sketch the cdf. 5. If X is a discrete random variable and you know the pdf f X of X, how can you find the cdf F X ? 6. If X is a discrete random variable and you know the cdf, how can you find the pdf? 7. If X is a random variable and a and b are real numbers, then it makes sense to talk about the random variable aX + b: It involves the same random process as X, but has values calculated as aX + b. Let Y = aX +b for some constants a and b. (Assume a ≠ 0.) i. If a > 0, express the cdf F Y (y) of Y in terms of the cdf F X of X. Show each step in your reasoning. ii. The same, but now assume a < 0. 8. If X is a continuous random variable and Y is as in Exercise 7, find the pdf f Y (y) of Y in terms of the pdf f X of X. [Hint: Exercises …