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Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. STA301 IDEA SOLUTIOON Created and written by Ali Question 1: Marks:2+3+3+3+4=15 a) If Z is a standard normal random variable with mean 0 and variance 1, then find the lower quartile. Answer; - Mean , µ = 0 Variance б2 = 1 Q1 = µ - 0.6745 б Q1 = 0 - .6745 (1) Q1 = -.6745 b) A random variable ‘x’ is said to be uniformly distributed such that c x k (i) Write down the density function of ‘x’ (ii) Prove that total area between the given limits is unity. Answer; - This is an example of this Question it is not correct answer; Probability Density Function The probability density function of a continuous random variable is a function which can be integrated to obtain the probability that the random variable takes a value in a given interval. More formally, the probability density function, f(x), of a continuous random variable X is the derivative of the cumulative distribution function F(x): Since it follows that: f (x )dx F (b ) F (a) p (a x b ) If f(x) is a probability density function then it must obey two conditions: a. that the total probability for all possible values of the continuous random variable X is 1: STA301 IDEA SOLUTIOON Created and written by Ali Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. STA301 IDEA SOLUTIOON Created and written by Ali b. that the probability density function can never be negative: f(x) > 0 for all x. (ii) Prove that total area between the given limits is unity. Answer; - c. the total probability for all possible values of the continuous random variable X is 1: c) What is the probability that a poker hand of 5 cards contain exactly 1 ace? Answer; - First, find the probability that the first card is an Ace and the 2nd to 5th are not: (4/52) * (48/51) * (47/50) * (46/49) * (45/48). Next, find the probability that the 2nd card is an Ace and the first, 3rd, 4th and 5th are not: (48/52) * (4/51) * (47/50) * (46/49) * (45/48). Next, the 3rd card is an Ace; the others are not: (48/52) * (47/51) * (4/50) * (46/49) * (45/48). Next, 4th card is an Ace; not the others: (48/52) * (47/51) * (46/50) * (4/49) * (45/48). Lastly, the 5th card is an Ace; not the others: (48/52) * (47/51) * (46/50) * (45/49) * (4/48). Adding these, the answer is: STA301 IDEA SOLUTIOON Created and written by Ali Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. STA301 IDEA SOLUTIOON Created and written by Ali 0.29947... which is just under 30%. d) A certain type of storage battery lasts on the average 3.0 years, with a standard deviation of 0.5 year. Assuming that the battery lives are normally distributed, find the probability that a given battery will last less than 2.3 years. Answer; - e) The incident of occupational disease is such that the workers have 20 percent chance of suffering from it. What is the probability that out of six more than 4 workers will come in contact of the disease? Answer; - P(4 < x <6) = 20 /100 = 0.2 Question 2: Marks:3+3+4+5=15 a) The Probability distribution of X is given below: Compute, i) P(x=2) Answer; - P(x = 2) = 35/100 = 0.35 ii) E(X) Answer; - E(X) = ∑(x) p(X) = 1.36 because (x) p(X) 0 0.5 0.7 0.12 0.04 ∑ (x) p(X) = 1.36 STA301 IDEA SOLUTIOON Created and written by Ali Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. STA301 IDEA SOLUTIOON Created and written by Ali X 0 1 2 3 4 1 1 35 4 1 P(X) 10 2 100 100 100 b) The p. d. f. of a random variable is given f(x) = (2x-1) 1<x < 2 0 Elsewhere Calculate P(X<1.5). Answer; - 1 1 (2x 1)dx 2x x 0 0 1 solve your self also i think it is the answer c) The given information is obtained from a joint distribution of X and Y. Calculate Correlation coefficient ( ) between X and Y. 21 34 49 33 86 EX , E Y , E XY E ( X 2 ) and E Y 2 15 15 15 15 15 Answer; - Cov (X ,Y ) xy V er (x ) V er ( y ) Cov(X,Y) = E(XY) - E(X) E(Y) = 3.26-(1.4)(2.26)=3.26-3.164=0.096 0.096 Now, 2 Var(x) = E (X )2 E (X ) 0.24 Var(y) = E (Y ) 2 [E (Y )]2 0.6 STA301 IDEA SOLUTIOON Created and written by Ali Generated by Foxit PDF Creator © Foxit Software http://www.foxitsoftware.com For evaluation only. STA301 IDEA SOLUTIOON Created and written by Ali 0.096 xy 0.25 0.24 x 0.6 d) Find the value of A, such that the function f(x, y) is a density function. f x, y Ay 1 x 0 x 1, 0 y 1 0 elsewhere Answer; - 1 1 ay (1 x )dxdy 0 0 1 1 1 a ( y (1 x )dx )dy 1 0 0 1 1 a ( ( y yx )dx )dy 1 0 0 1 1 x2 a yx y dy 1 0 2 0 after adding limit we get 1 y a y dy 1 0 2 1 y2 y2 A 1 2 4 0 2 1 A 1 2 1 A 1 4 A 4 i think it is the answer solve your self also STA301 IDEA SOLUTIOON Created and written by Ali

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STA301 Assignment No. 3 solution (thanks Ali, to share this with vusolutions)

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