If X is a random variable having a uniform distribution U (a,b) such that P(200<X<100)=0.5 and mean = 200, find a and b

Uniform distribution

For X∼ U(a,b)

f(x) = \$1/(b-a)\$ a<x<b

From the given data

P(200<X<100)=0.5,Mean=200

Substitute the given values in formula

\$\int_100^200 1/(b-a) dx\$ = 0.5

Simplification

\$(1/(b-a)) (x)_200^100\$ = 0.5

After simplification

\$1/(b-a) (200-100)\$ = 0.5

\$1/(b-a) 100\$ = 0.5

b-a = 200 ⇒equation(1)

Mean = 200

\$(a+b)/2 = 200 \$

\$a+b = 400\$

\$a = 400 - b\$ ⇒ equation(2)

Simplify equation(2) in (1) to get a,b values

b-(400 -b)=200

b-400+b=200

2b-400 = 200

2b = 200 + 400

2b = 600 ⇒ \$b = 300\$

Substitute b = 300 in equation(2)

a = 400 - 300

\$a=100\$

Uniform distribution

X ∼ U(100,300)

Answer

D