Answer: a. Yes.
c. 0.85
d. 0.60
e. 0.92
Step-by-step explanation:
Given discrete probability distribution :
x 15 22 34 40
P(X = x) 0.08 0.52 0.25 0.15
a. The sum of the all probabilities in an distribution is 1.
here, Total sum = P(X= 15) + P(X=22) + P(X=34) + P(X=40)
= 0.08 + 0.52+ 0.25 +0.15=1
∴ Yes , It is a valid probability distribution.
c. The probability that the random variable X is less than 39 : P(X<39)
= P(X= 15) + P(X=22) + P(X=34)
= 0.08 + 0.52+ 0.25 = 0.85
∴ The probability that the random variable X is less than 39 = 0.85
d. The probability that the random variable X is strictly between 9 and 24
:P( 9<X<24)= P(X= 15) + P(X=22)
= 0.08 + 0.52 =0.60
∴ The probability that the random variable X is strictly between 9 and 24=0.60
e. The probability that the random variable X is greater than 17 : P(X>17)
= P(X=22) + P(X=34) + P(X=40)
=0.52+ 0.25 +0.15=0.92
∴ The probability that the random variable X is greater than 17=0.92
Probabilities are used to determine the chance of events
Valid distribution
For a discrete probability distribution to be valid, the following must be true
[tex]\sum p(x)= 1[/tex]
So, we have:
[tex]\sum p(x) = P(X= 15) + P(X=22) + P(X=34) + P(X=40)[/tex]
Substitute known values
[tex]\sum p(x) = 0.08 + 0.52+ 0.25 +0.15[/tex]
[tex]\sum p(x) =1[/tex]
Hence, the discrete probability distribution is a valid probability distribution.
Probability that X is less than 39
This is represented as: P(x<39)
So, we have:
[tex]P(X<39)= P(X= 15) + P(X=22) + P(X=34)[/tex]
Substitute known values
[tex]P(X<39)= 0.08 + 0.52+ 0.25[/tex]
[tex]P(X<39)= 0.85[/tex]
Hence, the probability that the random variable X is less than 39 is 0.85
Probability that X is greater than 17
This is represented as: P(X>17)
So, we have:
[tex]P(X>17) = P(X=22) + P(X=34) + P(X=40)[/tex]
Substitute known values
[tex]P(X>17) =0.52+ 0.25 +0.15[/tex]
[tex]P(X>17) =0.92[/tex]
Hence, the probability that the random variable X is greater than 17 is 0.92
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