Download Unit-1-Important Questions 2 Marks

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.