Answer:
a) [tex] X \sim Unif(a= 1, b=5)[/tex]
The density function for this case is given by:
[tex] f(X) = \frac{1}{b-a}= \frac{1}{5-1}= \frac{1}{4} , 1\leq X \leq 5[/tex]
And the distribution is on the figure attached
b) [tex] P(X <1.5)[/tex]
And for this case we can use the cumulative distribution function given by:
[tex] F(x) = \frac{x-a}{b-a}= \frac{x-1}{5-1}= \frac{x-1}{4}, 1\leq X \leq 5[/tex]
And if we use this formula we got:
[tex]P(X<1.5) = \frac{1.5-1}{4}= \frac{0.5}{4}= \frac{1}{8}=0.125[/tex]
c) [tex] P(X >2.5) [/tex]
And we can use the complement rule and the cimulative distribution function and we can rewrite the expression like this:
[tex] P(X >2.5) = 1-P(X<2.5) = 1-F(2.5) = 1-\frac{2.5-1}{5-1}= 1-\frac{1.5}{4}= 1-\frac{3}{8}= \frac{5}{8}=0.625[/tex]
d) [tex]P(X<1) = \frac{1-1}{4}= \frac{0}{4}= 0[/tex]
Step-by-step explanation:
Part a
For this case we define the random variable X and the distribution for X is given by:
[tex] X \sim Unif(a= 1, b=5)[/tex]
The density function for this case is given by:
[tex] f(X) = \frac{1}{b-a}= \frac{1}{5-1}= \frac{1}{4} , 1\leq X \leq 5[/tex]
And the distribution is on the figure attached
Part b
For this case we want this probability:
[tex] P(X <1.5)[/tex]
And for this case we can use the cumulative distribution function given by:
[tex] F(x) = \frac{x-a}{b-a}= \frac{x-1}{5-1}= \frac{x-1}{4}, 1\leq X \leq 5[/tex]
And if we use this formula we got:
[tex]P(X<1.5) = \frac{1.5-1}{4}= \frac{0.5}{4}= \frac{1}{8}=0.125[/tex]
Part c
For this case we want this probability:
[tex] P(X >2.5) [/tex]
And we can use the complement rule and the cimulative distribution function and we can rewrite the expression like this:
[tex] P(X >2.5) = 1-P(X<2.5) = 1-F(2.5) = 1-\frac{2.5-1}{5-1}= 1-\frac{1.5}{4}= 1-\frac{3}{8}= \frac{5}{8}=0.625[/tex]
Part d" What is the probability that X lies below 1?
[tex] P(X <1)[/tex]
And for this case we can use the cumulative distribution function given by:
[tex] F(x) = \frac{x-a}{b-a}= \frac{x-1}{5-1}= \frac{x-1}{4}, 1\leq X \leq 5[/tex]
And if we use this formula we got:
[tex]P(X<1) = \frac{1-1}{4}= \frac{0}{4}= 0[/tex]
