With diagram -16 Let the interval |-r.r]be the base of a semicircle. Ifa point isselected at...
Let the interval [-r,r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2. Answer this by using Pythagorean theorem and explain all of the steps well and clearly.
Let the interval [-r,r] be the base of a semicircle. If a point is selected at random from this interval, assign a probability to the event that the length of the perpendicular segment from the point to the semicircle is less than r/2. Answer this by using Pythagorean theorem and explain all of the steps well and clearly.
are even, r even. A 1.1-14·Let the interval [-r,r] be the base of a semicircle. ne of the f a pot is selected at random from this interval, assign abilty of a probability to the event that the length of the perpen- e slot into dicular segment from the point to the semicircle is less than r/2. 1.1-15. Let S = A1 U A2 U U Am, where events (a) If P(A1) P(A2)P(A), show that P(Ai - (b) If A...
Let C be the semicircle of radius r > 0 with center at (0, 0) and lying above the x-axis. For each x in [−r, r], let L(x) be the length of the line from (x, 0) to the semicircle C and perpendicular to the x-axis. What is the probability that L(x) is less than r/2?
3. Let X be a continuous random variable defined on the interval 0, 4] with probability density function p(r) e(1 +4) (a) Find the value of c such that p(x) is a valid probability density function b) Find the probability that X is greater than 3 (c) If X is greater than 1, find the probability X is greater than 2 d) What is the probability that X is less than some number a, assuing 0<a<4?
Suppose 6 numbers are generated by a computer, each uniform on the interval (0, 1). Let Y be the random variable representing the smallest of the numbers. (a) Show that the probability density of Y is given by py (t) -61-t)5, 0t <1 [51 Hint: The probability density for the r-th largest random variable can be derived using the Beta distribution by letting a = r and ?-n-r +1. (b) What is the probability that the smallest number is less...
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4 3 marks (b) Suppose X, and X2 are two iid normal N(μ, σ2) variables. Define Are random variables V and W independent? Mathematically justify your answer. 3 marks] (c) Let C denote the unit circle...
4. Let X be a continuous random variable defined on the interval [1, 10 with probability density function r2 (a) Find the value of c such that p(x) is a valid probability density function. (b) Find the probability that X is larger than 8 or less than 2 (this should be one number! (c) Find the probability that X is larger than some value a, assuming 1 < a< 10 d) Find the probability that X is more than 3
1. (a) A point is selected at random on the unit interval, dividing it into two pieces with total length 1. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than 1/4. 3 marks (b) Suppose X1 and X2 are two iid normal N(μ, σ*) variables. Define Are random variables V and W independent? Mathematically justify your answer 3 marks (c) Let C denote the unit circle...
Let X be a continuous random variable defined on the interval [0, 4] with probability density function p(x) = c(1 + 4x) (a) Find the value of c such that p(x) is a valid probability density function. (b) Find the probability that X is greater than 3. (c) If X is greater than 1, find the probability X is greater than 2. (d) What is the probability that X is less than some number a, assuming 0 < a <...