Questions: Question 16 (1 point) Find the expected value for the random variable x having this probability function. a=19 b=20 c=21 d=22 e=23 a 17 b 17.2 c 21 d 16.4

Question 16 (1 point) Find the expected value for the random variable x having this probability function. a=19 b=20 c=21 d=22 e=23 a 17 b 17.2 c 21 d 16.4

Transcript text: Question 16 (1 point) Find the expected value for the random variable x having this probability function. \[ \begin{array}{lll} a=19 & b=20 & c=21 \\ d=22 & e=23 & \end{array} \] a $\quad 17$ b $\quad 17.2$ c $\quad 21$ d $\quad 16.4$

Solution

Solution Steps

Step 1: Determine the probability for each value of x.

The graph represents a uniform distribution where the probability is constant for all values of x between a and e. The probability is given by the height of the rectangle, which is 2. The width of each interval is 1. Since the area of the rectangle represents the total probability, which is 1, we have:

2 * (e - a) = 1, where (e-a) represents the width of the interval from a to e. Since we have a uniform distribution, the probability for each interval is 1/(e - a). Since we are given a = 19 and e = 23, the number of possible outcomes is e - a = 23 - 19 = 4.

The probability for each value is: P(x=19) = 1/4 P(x=20) = 1/4 P(x=21) = 1/4 P(x=22) = 1/4 P(x=23) = 1/4

Step 2: Calculate the expected value.

The expected value E(x) is the sum of each value of x multiplied by its probability.

E(x) = 19 * (1/4) + 20 * (1/4) + 21 * (1/4) + 22 * (1/4) + 23 * (1/4) E(x) = (1/4) * (19 + 20 + 21 + 22 + 23) E(x) = (1/4) * 105 E(x) = 21.25