Question

1 & 2 pls

Let U be a uniformly distributed random variable on [0, 1]. What is the probability that the equation x2 + 4-U、x + 1 = 0 has two distinct real roots x1 and x2? 1. 2. The probability that an electron is at a distance r from the center of the nucleus is: with R being a scale constant. a) Find the value of the constant C. b) Find the mean radius f. c) Find the standard deviation of the distance distribution.

Answer 1

U elementof [0,1] x^2 + 4Ux + 1 = 0 at U = 1, x^2 + 4x = 1 = 0 x^2 + 4x + 1 = 0 x = -2 plusminus squareroot 3 real both at U = 1/2, x^2 + 2x + 1 = 0 (x + 1)^2 = 0 x = -1, -1 real and same at U = 1/4, x^2 + x + 1 = 0 x = -1 plusminus squareroot 3i/2 complex so at U = 0, x^2 = -1 x = i, i imaginary so in range U elementof [1/2 1] rightarrow real roots and U elementof [0, 1/2] rightarrow Imaginary or complex roots so probably of two distinct real roots is rho = 1/2 because 1/2 region cover for real roots in [0 1] P9r) = c middot lambda^2 e^-r/R integral6 infinity_0 P(r) dr = 1 \r\n integral^infinity_0 c r^2 e^-r/R dr = 1 C(2!)/(1/R^3) = 1 C = (1/R^3)1/2 C = 1/2R^3 r
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