# Probability And Statistics | Week 7

## Probability And Statistics Week 7 Answers

**Link : Probability And Statistics (nptel.ac.in)**

**Q1. Let X₁, X₁, X3 and X4 be independent random variables with respective means 1,1/2,1/3,1/4. Then Y = min(X₁, X₂, X3, X4) has an exponential distribution with the mean equal to****(A) 0.1**

(B) 0.48

(C) 2.08

(D) 0.5

**Q2. Test results from an electronic circuit board indicate that 50% of board failures are caused by assembly defects, 40% are due to electrical components and 10% are due to mechanical defects. Suppose that 10 boards fail independently. Let the random variables X, Y and Z denote the number of assembly, electrical and mechanical defects among the 10 boards. Calculate P(X= 5, Y = 3, Z = 2).**

(A) 0.0602**(B) 0.0504**

(C) 0.0204

(D) 0.0806

**Q3. Soft drink cans are filled by an automated filling machine. Assume that the fill volumes of the cans are independent, normal random variables with the standard deviation 15mL. Suppose the probability that the mean of a sample of 100 cans is below 350mL is 0.005. What should the mean fill volume be equal to?**

(A) 258.369**(B) 353.8637**

(C) 360.257

(D) 257.952

**Q4. A plastic casing for a magnetic disk is composed of two halves. The thickness of each half is normally distributed with a mean of 2 millimeters and a standard deviation of 0.1 millimeter and the two halves are independent. What is the probability that the total thickness exceeds 4.3 millimeters?****(A) 0.01695**

(B) 0.04962

(C) 0.02032

(D) 0.07081

**Q5. The joint density function of continuous random variables X and Y is given byf(x,y) = {x + y, 0<x< 1,0 <y<1 0, otherwiseLet U=X-Y. Find Var (U)**

(A) 1/4

**(B) 1/6**

(C) 1/18

(D) 1/3

**Q6. Suppose that X₁, X₂, …, Xn are random variables such that the variances of each variable are 1 and correlation between each pair of different variables is 1/4. Then Var (X₁ + X₂ + … + Xn) is?**

(A) n(n+1)/2

(B) n(n+2)/4**(C) n(n+3)/4**

(D) n(n+3)/2

**Q7. The joint distribution of the continuous random variables X, Y and Z is constant over the region x² + y² ≤ 1,0 <z < 4. Determine marginal probability density function of X.**

(A) 8x²,0 < x < 1**(B) π²√1 − x²,−1≤ x ≤ 1**

(C) √1-x², -1 ≤ x ≤ 1

(D) x/2,-1/2≤x≤1/2

**Q8. X and Y be continuous random variables with the joint pdff _{XY} = {e^{-y}(1-e^{-x}), 0<x<y<∞ e^{-x}(1-e^{-y}), 0<y≤x<∞Find E(X+Y).**

(A) 1

(B) 2

**(C) 4**

(D) 6

**Q9. Suppose X₁,…, X10 are iid random variables following an exponential distribution with the mean 9. Find P(Y> 7|Y > 4), where Y = min {X₁,…,X _{10}}**

**(A) 0.03579**

(B) 0.09697

(C) 0.18888

(D) 0.01312

**Q10. Let X₁, X2, X3, X4 and X5 be iid with common pdf given byf(x) = { 1, 0<x< 1 0, otherwiseSuppose pdf of X(3) is of the form g3 (y3) = Ay**

_{3}

^{B}(1 – y

_{3}); 0 < y

_{3}< 1. Then (A, B, C) is

**(A) (30,2,2)**

(B) (30,2,3)

(C) (15,2,2)

(D) (30,4,3)

anybody tell me answer of 6 clearly!