Answer:
(1) (c) <u>5.30 years</u>.
(2) (b) <u>0.289</u>.
(3) (b) <u>0.80</u>.
(4) (d) <u>0.50</u>.
(5) (a) <u>5.25 years</u>.
Step-by-step explanation:
Let <em>X</em> = age of the children in kindergarten on the first day of school.
The random variable <em>X</em> follows a continuous Uniform distribution with parameters <em>a</em> = 4.8 years and <em>b</em> = 5.8 years.
The probability density function function of <em>X</em> is:
![f_{X}(x)=\left \{ {{\frac{1}{b-a}} ;\ a](https://tex.z-dn.net/?f=f_%7BX%7D%28x%29%3D%5Cleft%20%5C%7B%20%7B%7B%5Cfrac%7B1%7D%7Bb-a%7D%7D%20%3B%5C%20a%3CX%3Cb%2C%5C%20a%3Cb%5Catop%20%7B0%3B%5C%20otherwise%7D%7D%20%5Cright.)
(1)
The expected value of a Uniform random variable is:
![E(X)=\frac{1}{2}(a+b)](https://tex.z-dn.net/?f=E%28X%29%3D%5Cfrac%7B1%7D%7B2%7D%28a%2Bb%29)
Compute the mean of <em>X</em> as follows:
![E(X)=\frac{1}{2}(a+b)=\frac{1}{2}\times (4.8+5.8)=5.3](https://tex.z-dn.net/?f=E%28X%29%3D%5Cfrac%7B1%7D%7B2%7D%28a%2Bb%29%3D%5Cfrac%7B1%7D%7B2%7D%5Ctimes%20%284.8%2B5.8%29%3D5.3)
Thus, the mean of the distribution is (c) <u>5.30 years</u>.
(2)
The standard deviation of a Uniform random variable is:
![SD(X)=\sqrt{\frac{1}{12}(b-a)^{2}}](https://tex.z-dn.net/?f=SD%28X%29%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B12%7D%28b-a%29%5E%7B2%7D%7D)
Compute the standard deviation of <em>X</em> as follows:
![SD(X)=\sqrt{\frac{1}{12}(b-a)^{2}}=\sqrt{\frac{1}{12}\times (5.8-4.8)^{2}}=0.289](https://tex.z-dn.net/?f=SD%28X%29%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B12%7D%28b-a%29%5E%7B2%7D%7D%3D%5Csqrt%7B%5Cfrac%7B1%7D%7B12%7D%5Ctimes%20%285.8-4.8%29%5E%7B2%7D%7D%3D0.289)
Thus, the standard deviation of the distribution is (b) <u>0.289</u>.
(3)
Compute the probability that a randomly selected child is older than 5 years old as follows:
![P(X>5)=\int\limits^{5.8}_{5} {\frac{1}{5.8-4.8}}\, dx\\](https://tex.z-dn.net/?f=P%28X%3E5%29%3D%5Cint%5Climits%5E%7B5.8%7D_%7B5%7D%20%7B%5Cfrac%7B1%7D%7B5.8-4.8%7D%7D%5C%2C%20dx%5C%5C)
![=\int\limits^{5.8}_{5} {1}\, dx\\=[x]^{5.8}_{5}\\=(5.8-5)\\=0.8](https://tex.z-dn.net/?f=%3D%5Cint%5Climits%5E%7B5.8%7D_%7B5%7D%20%7B1%7D%5C%2C%20dx%5C%5C%3D%5Bx%5D%5E%7B5.8%7D_%7B5%7D%5C%5C%3D%285.8-5%29%5C%5C%3D0.8)
Thus, the probability that a randomly selected child is older than 5 years old is (b) <u>0.80</u>.
(4)
Compute the probability that a randomly selected child is between 5.2 years and 5.7 years old as follows:
![P(5.2](https://tex.z-dn.net/?f=P%285.2%3CX%3C5.7%29%3D%5Cint%5Climits%5E%7B5.7%7D_%7B5.2%7D%20%7B%5Cfrac%7B1%7D%7B5.8-4.8%7D%7D%5C%2C%20dx%5C%5C)
![=\int\limits^{5.7}_{5.2} {1}\, dx\\=[x]^{5.7}_{5.2}\\=(5.7-5.2)\\=0.5](https://tex.z-dn.net/?f=%3D%5Cint%5Climits%5E%7B5.7%7D_%7B5.2%7D%20%7B1%7D%5C%2C%20dx%5C%5C%3D%5Bx%5D%5E%7B5.7%7D_%7B5.2%7D%5C%5C%3D%285.7-5.2%29%5C%5C%3D0.5)
Thus, the probability that a randomly selected child is between 5.2 years and 5.7 years old is (d) <u>0.50</u>.
(5)
It is provided that a randomly selected child is at the 45th percentile.
This implies that:
P (X < x) = 0.45
Compute the value of <em>x</em> as follows:
![P (X < x) = 0.45](https://tex.z-dn.net/?f=P%20%28X%20%3C%20x%29%20%3D%200.45)
![\int\limits^{x}_{4.8} {\frac{1}{5.8-4.8}}\, dx=0.45](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%7Bx%7D_%7B4.8%7D%20%7B%5Cfrac%7B1%7D%7B5.8-4.8%7D%7D%5C%2C%20dx%3D0.45)
![\int\limits^{x}_{4.8} {1}\, dx=0.45](https://tex.z-dn.net/?f=%5Cint%5Climits%5E%7Bx%7D_%7B4.8%7D%20%7B1%7D%5C%2C%20dx%3D0.45)
![[x]^{x}_{4.8}=0.45](https://tex.z-dn.net/?f=%5Bx%5D%5E%7Bx%7D_%7B4.8%7D%3D0.45)
![x-4.8=0.45\\](https://tex.z-dn.net/?f=x-4.8%3D0.45%5C%5C)
![x=0.45+4.8\\x=5.25](https://tex.z-dn.net/?f=x%3D0.45%2B4.8%5C%5Cx%3D5.25)
Thus, the age of the child at the 45th percentile is (a) <u>5.25 years</u>.