Which of the following tables shows a valid probability density function? Select all correct answers. Select all that apply:   x P(X=x) 0 14 1 12 2 18 3 18     x P(X=x) 0 35 1 15 2 −15 3 25     x P(X=x) 0 310 1 15 2 35 3 110     x P(X=x) 0 0.01 1 0.46 2 0.19     x P(X=x) 0 0.09 1 0.24 2 0.67     x P(X=x) 0 14 1 12 2 14

Determine whether the following function is a probability density function. f(x) = x, 0sxs8 32 Is the function f(x) a probability density function? No Yes

Q2. The probability density function of X is as follows. f(x) = 2 - 2x, for x = [0, 1] Which one of the following expressions can be used to find P(0.5 < X < 1.5)? F(x) is the cumulative distribution function of X. (A) 1 - F(0.5) (C) F (1.5)-f(0.5) (B) f(1.5)-f(0.5) (D) 1.5 (2 - 2x) dx (E) other

Given the probability distributions for variables X and Y shown to the right, compute the terms below a. E(X) and E(Y) b. ox and dy C. OXY d. E(X+Y) P(X,Y) 0.4 0.6 X 100 300 Y 300 100 0

Suppose a random variable X has a probability density function in img.jpg.  a. Find the value of k that makes this a probability density function. b. Find P(0.4≤X≤1). c. Find P(X≤0.4|X≤0.8) d. Find F(b) = P(X≤b) and sketch the graph of this function.

Find k such that the function is a probability density function over the given interval. Then write the probability density function. f(x) = kx, 1≤x≤8 What is the value of k? k = 1 (Simplify your answer.) What is the probability density function? f(x) =

Let X be the total amount saved in dollars at the cash register. The pdf of X is given by: Value of X 20 28 40 50 58 70 probability 0.36 0.21 0.24 0.1225 0.0175 0.03 .0175 0.0025 Before I make the purchase, you would place my chance of saving $20 at 24%. Suppose you approach me after I pay for my items and ask: "Did you save any money?" If I answer "no", then you know I didn't save $20 and thus the chance of saving $20 changes from the unconditional chance 24% to the conditional chance 0%. However, if I answer "yes" my chance of saving $20 would get higher. What would be the conditional chance of saving $20 if you knew I got a discount? Mathematically, we seek: P(X = 20|X > 0 ). Even though it looks weird, it is just a conditional probability. Find it. P(X = 20|X > 0) = P(X = 20 and X > 0) / P(X > 0) = Hint: P(X = 20 and X > 0) be read off the table by identifying possible values where the event (X = 20 and X > 0) is true, don't use the multiplication rule. Do not round, express your…