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- The probability for a continuous variable that is normally distributed is equivalent to what?9. Suppose the distance X between a point target and a shot aimed at a point in a coin operated target game is continuous random variable with probability density function a. b. C. d. f(x) = (1 − x²) 0 -1≤x≤1 otherwise Sketch the graph of f(x) compute P(X>0) compute p(-1/2 ¹14)Q3. Exponential Distribution has a memoryless property. Intuitively, it means that the probability of customer service answering you call (assuming waiting time is exponential) in the next 10 mins is the same, no matter if you have waited an hour on the line or just picked up the phone. Formally, if X ∼ exponential(λ), f(x) = λ exp(- λx), and t and s are two positive numbers, use the definition of conditional probability to show that P(X > t + s | X > t) = P(X > s).Hint: Find the cdf of X first, and note that P(X > t + s Ç X > t) = P(X > t + s)
- 2. The probability in minutes of being waited on in a large chain restaurant is given by the frequency function f(t) = t³;0< t< 3. What is the probability of being waited on 81 between 1 and 2 minutes?Using 2 independent DRVS X and Y with probability distribution px(x) and py (y) below, find E(Z) where Z = X + Y X 1 2 3 4 Px(x) 1/8 3/16 3/8 3/16 1/8 Y 3 4 5 Py(y) 1/4 1/4 1/4 1/4If F(x) is the distribution function of X given by F ( x ) = 0 if x ≤ 1 k ( x − 1 )4 if 1 3 Colon noiludleb sdt Determine (i) f ( x ) (ii) k
- Q3. Exponential Distribution has a memoryless property. Intuitively, it means that the probability of customer service answering you call (assuming waiting time is exponential) in the next 10 mins is the same, no matter if you have waited an hour on the line or just picked up the phone. Formally, if X ~ exponential(X), f(x) = A exp(- Ax), and t and s are two positive numbers, use the definition of conditional probability to show that P(X > t +s| X > t) = P(X > s). Hint: Find the cdf of X first, and note that P(X > t +snX> t) = P(X > t + s)5.2.1 The conditional probability distribution of Y given X = x is fy.0) = xe-y for y> 0, and the marginal probability distribution of X is a continuous uniform distribution over 0 to 10. a. Graph fyv) = xe¬w for y > 0 for several values of x. Determine: b. P(Y < 2|X = 2) d. E(Y |X = x) f. fr(y) c. E(Y | X = 2) e. fxy(x, y)
