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Suppose that the cumulative distribution function of the random variable

aflacatn

aflacatn

Answered question

2021-10-17

Assuming that the random variable x has a cumulative distribution function,
F(x)={0,x<00.25x,0≤x<51,5≤x 
Determine the following:
a) p(x<2.8)
b) p(x>1.5)
c) p(x<−z)
d) p(x>b)

Answer & Explanation

Adnaan Franks

Adnaan Franks

Skilled2021-10-18Added 92 answers

Step 1
So we have,
The cumulative distribution function of the random variable x is.
F(x)={0,x<00.25x,0≤x<51,x≥5 
Step 2
Determine P(X<2.8)
That is,
P(X<2.8)=F(2.8)
=0.25(2.8)
=0.7
Step 3
Determine P(X>1.5)
That is,
P(X>1.5)=1−P(X≤1.5)
=1−F(1.5)
=1−[0.25(1.5)]
=1−0.375
=0.625
Step 4
Determine P(X<−2)
That is,
P(X<−2)=F(−2)
=0
Step 5
Determine P(X>6)
That is,
P(X>6)=1−P(X≤6)
=1−F(6)
=1−1
=0

Andre BalkonE

Andre BalkonE

Skilled2023-06-19Added 110 answers

Step 1:
a) To find P(x<2.8), we need to evaluate the cumulative distribution function (CDF) at x=2.8. The CDF, denoted as F(x), gives the probability that the random variable x takes on a value less than or equal to x. In this case, we have:
F(x)={0,x<00.25x,0≤x<51,5≤x
Since 2.8 falls within the interval 0≤x<5, we can use the second case of the CDF. Thus,
P(x<2.8)=F(2.8)=0.25·2.8=0.7.
Therefore, the probability that x is less than 2.8 is 0.7.
Step 2:
b) To determine P(x>1.5), we need to find the probability that x is greater than 1.5. We can approach this by calculating the complement of P(x<1.5). Since 1.5 falls within the interval 0≤x<5, we can use the second case of the CDF. Thus,
P(x<1.5)=F(1.5)=0.25·1.5=0.375.
The complement of this probability is:
P(x>1.5)=1−P(x<1.5)=1−0.375=0.625.
Therefore, the probability that x is greater than 1.5 is 0.625.
Step 3:
c) The probability P(x<−z) represents the probability that x is less than a negative value −z. However, the given cumulative distribution function has a probability of 0 for all negative values of x, as stated in the first case of the CDF. Therefore, P(x<−z)=0 for any negative value −z.
Step 4:
d) Similarly, the probability P(x>b) represents the probability that x is greater than a given value b. In this case, since the given cumulative distribution function has a probability of 1 for all values greater than or equal to 5, we have:
P(x>b)=1 for any value b.
Jazz Frenia

Jazz Frenia

Skilled2023-06-19Added 106 answers

a) P(X<2.8):
Since the cumulative distribution function (CDF) is defined piecewise, we can evaluate the probability by subtracting the CDF values at the endpoints:
P(X<2.8)=F(2.8)−F(0)=0.25(2.8)−0=0.7
b) P(X>1.5):
To find the probability that X is greater than 1.5, we subtract the CDF value at 1.5 from 1 (since the CDF gives the probability of being less than or equal to a certain value):
P(X>1.5)=1−F(1.5)=1−0.25(1.5)=1−0.375=0.625
c) P(X<−z):
Since the CDF is 0 for negative values, the probability of X being less than −z is always 0:
P(X<−z)=0
d) P(X>b):
Given a value b greater than 5, the probability that X is greater than b is 0 since the CDF is 1 for x≥5:
P(X>b)=0
fudzisako

fudzisako

Skilled2023-06-19Added 105 answers

Answer:
a) P(x<2.8)=0.25(2.8)
b) P(x>1.5)=1−0.25(1.5)
c) P(x<−z)=0 (for any value of z)
d) If b≥5, P(x>b)=0. Otherwise, P(x>b)=1−0.25b.
Explanation:
F(x)={0,x<00.25x,0≤x<51,5≤x
a) To find P(x<2.8), we can evaluate the CDF at x=2.8:
P(x<2.8)=F(2.8)
Since 0≤x<5 for 2.8, we use the second case of the CDF:
P(x<2.8)=F(2.8)=0.25(2.8)
b) To find P(x>1.5), we can use the complement rule. The probability of an event not occurring is equal to one minus the probability of it occurring. Therefore:
P(x>1.5)=1−P(x≤1.5)
To find P(x≤1.5), we evaluate the CDF at x=1.5:
P(x≤1.5)=F(1.5)
Since 0≤x<5 for 1.5, we use the second case of the CDF:
P(x≤1.5)=F(1.5)=0.25(1.5)
Therefore, P(x>1.5)=1−0.25(1.5)
c) To find P(x<−z), we need to consider the definition of the CDF for x<0. Since the CDF is zero for x<0, it implies that the probability of x being less than any negative value is also zero. Therefore, P(x<−z)=0 for any value of z.
d) To find P(x>b), we can use the complement rule. The probability of x being greater than b is equal to one minus the probability of x being less than or equal to b. Therefore:
P(x>b)=1−P(x≤b)
To find P(x≤b), we evaluate the CDF at x=b.
If b≥5, then P(x≤b)=F(b)=1, and therefore P(x>b)=1−1=0.
If 0≤b<5, then P(x≤b)=F(b)=0.25b, and therefore P(x>b)=1−0.25b.

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