Let x be the number of courses for which a randomly selected student at a certain university is registered. The probability distribution of x appears in the table shown below:

x 1 2 3 4 5 6 7
p(x) .04 .05 .08 .26 .39 .16 .02
(a) What is P(x = 4)?
P(x = 4) =

(b) What is P(x 4)?
P(x 4) =

(c) What is the probability that the selected student is taking at most five courses?
P(at most 5 courses) =

(d) What is the probability that the selected student is taking at least five courses? more than five courses?
P(at least 5 courses) =
P(more than 5 courses) =

(e) Calculate P(3 x 6) and P(3 < x < 6).
P(3 x 6) =
P(3 < x < 6) =

Given probability distribution (pdf):

x 1 2 3 4 5 6 7
p(x) .04 .05 .08 .26 .39 .16 .02

Since &Sum;p(x) for x=1 to 7 =1.0
we conclude that:
p(0)=0
p(1)=0.04
...
p(7)=0.02
p(8+)=0
(a)
P(x=4)=0.26/1=0.26
(b)
P(x>4)
=P(5)+P(6)+P(7)+P(8+)
=0.39+0.16+0.02+0
=0.53
(c)
P(x≤5)=P(0-)+P(1)+P(2)+P(3)+P(4)+P(5)
=?

(d) P(x≥5)=P(x>4)

(e)
P(3≤x≤6)=P(3)+P(4)+P(5)+P(6)
P(3<x<6)=P(4)+P(5)

Fifty students are polled on the courses they are taking, and X is the number of courses a student takes. The number of courses ranges from 2 to 6. Is the following a probability distribution of X?

(a) P(x = 4) = .26

(b) P(x < 4) = .04 + .05 + .08 = .17

(c) P(at most 5 courses) = P(x ≤ 5) = .04 + .05 + .08 + .26 + .39 = .82

(d) P(at least 5 courses) = P(x ≥ 5) = .39 + .16 + .02 = .57
P(more than 5 courses) = P(x > 5) = .16 + .02 = .18

(e) P(3 ≤ x ≤ 6) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) = .08 + .26 + .39 + .16 = .89
P(3 < x < 6) = P(x = 4) + P(x = 5) = .26 + .39 = .65

(a) P(x = 4) = 0.26

(b) P(x ≥ 4) = P(x = 4) + P(x = 5) + P(x = 6) + P(x = 7) = 0.26 + 0.39 + 0.16 + 0.02 = 0.83

(c) P(at most 5 courses) = P(x ≤ 5) = P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) = 0.04 + 0.05 + 0.08 + 0.26 + 0.39 = 0.82

(d) P(at least 5 courses) = P(x ≥ 5) = P(x = 5) + P(x = 6) + P(x = 7) = 0.39 + 0.16 + 0.02 = 0.57
P(more than 5 courses) = P(x > 5) = P(x = 6) + P(x = 7) = 0.16 + 0.02 = 0.18

(e) P(3 < x < 6) = P(x = 4) + P(x = 5) = 0.26 + 0.39 = 0.65
P(3 ≤ x ≤ 6) = P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6) = 0.08 + 0.26 + 0.39 + 0.16 = 0.89

(a) To find P(x = 4), we can look at the given probability distribution table. The value of P(x = 4) is given as 0.26.

(b) P(x > 4) represents the probability that the number of courses is greater than 4. To find this, we sum the probabilities of x values greater than 4. From the table, the probabilities for x = 5, 6, and 7 are 0.39, 0.16, and 0.02 respectively. Adding these probabilities, we get: P(x > 4) = 0.39 + 0.16 + 0.02 = 0.57.

(c) To find the probability that the selected student is taking at most five courses, we sum the probabilities for x values less than or equal to 5. From the table, the probabilities for x = 1, 2, 3, 4, and 5 are 0.04, 0.05, 0.08, 0.26, and 0.39 respectively. Adding these probabilities, we get: P(at most 5 courses) = 0.04 + 0.05 + 0.08 + 0.26 + 0.39 = 0.82.

(d) To find the probability that the selected student is taking at least five courses, we can subtract the probability of taking at most four courses from 1. From part (c), we know that P(at most 5 courses) = 0.82. Therefore, P(at least 5 courses) = 1 - P(at most 5 courses) = 1 - 0.82 = 0.18.

To find the probability of taking more than five courses, we need to consider the probabilities of x values greater than 5. From the table, the probabilities for x = 6 and 7 are 0.16 and 0.02 respectively. Adding these probabilities, we get: P(more than 5 courses) = 0.16 + 0.02 = 0.18.

(e) P(3 x 6) represents the probability that the selected student is taking between 3 and 6 courses, inclusive. To calculate this, we sum the probabilities of x values from 3 to 6. From the table, the probabilities for x = 3, 4, 5, and 6 are 0.08, 0.26, 0.39, and 0.16 respectively. Adding these probabilities, we get: P(3 x 6) = 0.08 + 0.26 + 0.39 + 0.16 = 0.89.

P(3 < x < 6) represents the probability that the selected student is taking more than 3 but less than 6 courses. To calculate this, we exclude the probabilities of x values of 3 and 6. So, we consider the probabilities for x = 4 and 5. From the table, the probabilities for x = 4 and 5 are 0.26 and 0.39 respectively. Adding these probabilities, we get: P(3 < x < 6) = 0.26 + 0.39 = 0.65.