Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Unit-1-Important Questions 2 Marks: 1.Define Probability in different ways. 2.Define Total Probability & Baye’s Theorems? 3.What are the conditions for a function to be a Random variable? 4. Define (i) Binomial (ii) Exponential (iii) Uniform (iv) Rayleigh Functions? 5. Define Conditional Distribution and Density Functions? 10 Marks: 1. In three boxes there are capacitors as shown in below table. An experiment consists of first randomly selecting a box, assuming each box is likely and then selecting a capacitor from the chosen box. (a). What is the Probability of selecting a 0.01𝜇F capacitor given that box 2 is selected. (b). If 0.01𝜇F capacitor is selected what is the probability that it comes from box3. Capacitors (𝝁F) 0.01 0.1 1.0 Total Boxes 1 2 3 20 55 70 145 95 35 80 210 Total 25 75 145 245 140 165 295 600 2. A pair of dice is rolled 10 times. Find (a) The probability that number 7 will show at least once, (b) The probability that number 7 will show at least twice. 3. A random variable X has probabilities as shown in below table. (a) Find the value of K. (b) Find FX(X), fx(x) and draw the plots. -3 -2 -1 0 1 2 X P(X) 0.2 0.5K K 0.1 0.3K K 2 𝑥𝑒 −𝑥 /2 𝑓𝑜𝑟 𝑥 ≥ 0 4. A Rayleigh density function is given by fx(x) = { 0 𝑓𝑜𝑟 𝑥 < 0 ∞ (a) Prove that fx(x) satisfies the properties (i) fx(x) ≥0 for all x and (ii) ∫−∞ fx(x)dx = 1. (b)Find the Distribution function FX(X). (c) Find P (0.5≤ 𝑋 ≤ 2). (d) Find P (0.5<X<2). 5. Consider the probability density function fx(x) = a 𝑒 −𝑏|𝑥| , where x is a random variable whose allowable values range from -∞ to ∞. Find (i) CDF (ii) Relation between a and b. (iii) The probability that outcome x lies between 1 and 2.