Question

Let X be a random variable of the mixed type having the distribution function

F(x)=0wherex<0

F(x)=x24where0≤x<1

F(x)=x+14where1≤x<2

F(x)=1wherex≥2

Question 1: Find the mean of X

Question 2: Find the variance of X

Question 3: Find P(1/4 < X < 1)

Question 4: Find P(X = 1)

Question 5: Find P(X = 1/2)

Helpful Hint:

Find the F(x)=ddxP′(x)

F(x) = 0 when x < 0

F(x) = 2x4when 0 < x < 1

F(x) = 14when 1 < x < 2

F(x) = 1 when x > 2

Question 1: Mean of x

= 2∫01x24dx+x14+∫12x4dx+2x4=24[x34]01+14+14[x24]12+12=

Question 2: Variance of x

Will need to find:E(x2)=2∫01x2x4dx+x14+∫12x2x4dx+22x4=24[x44]01+14+14[x33]12+1=

Now Find V(X)

V(x)=E(x2)−(E(x))2

Question 3: P(1/4 < X < 1) = F(1′)−F(1/4) =

Question 4: P(X = 1) = F(1)−F(1′)

Question 5: P(X = 1/2) =

Answer 1