Question

Probability & Statistics (25 points) 1. (5 points) If the probability that student A will fail a certain statistics examination is 0.3, the probability that student B will fail the examination is 0.2, and the probability that bosh student A and student B will fail the examination is 0.1. a) What is the probability that at least one of these two students will fail the examination? b) What is the probability that exactly one of the two students will fail the examination? 2. (5 points) Suppose that the c.d.f. of a random variable X is as follows: for x 50 F(x) for 0<xs 2, for x > 2 Find and sketch the p.d.f. of X. 103-41 3. (5 points) If X has the uniform distribution on an interval [1, 3), then the p.d.f of X is as follows: for Isxs3 otherwise a) Find and sketch the c.d.f. of X. b) Determine the value of the probability: Pr (X<2). 4. (5 points) Suppose that the p.d.f of a random variable X is as follows: for Os x 52. otherwise a) Find the value of the constant c. b) Find the value of Pr (x<3/2). 5. (5 points) Suppose that X has the uniform distribution on the interval [0, 1]. Compute the expectation and variance of X

Answer 1

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(3 f(x) = f / 2 IL x 3 otherwise Х a. CDF Fx cx) 2 ] 2 dl Ex (a) XC- 's lans 3. os EXCX) 0 À x <1 uc I x3 2 . 1 x > 3 ܝ Exca 0.5 +* 3 3.5. 0.5 1 105 2 2.5 be por cx <] Ex (2) =0.5 22-1 2 =1 Q 2 2 @ f(x) dx = 1 - of x² dx 2 = ca ng C18-0]s 3 0 >> = 3/8 372 312 P[x <3 - 3 n² dx zxx² 2 8 8 3 Jo Scanhed with CamScanner =1[97 ) 27 64 8 8

Ques 50 XN Uniform (0, 1) So, fx (r) ; osal otherwise O for lluiform, ECX) atb et = 0.5 2 2 V(X) = (b-a)? = 0.0833 12 12 es Scanned with CamScanner