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26M1/STA21M1 Assignment Marks 50 Due date: 31 March 2023 1. Let X be the sum of the up faces on a roll of a pair of fair 6-sided dice. a. Determine the pmf of X. (5) b. Suppose A = {x : x = 7, 11} and B = {x : x = 2, 3, 12}. Then, using the values of f(x) in (a), calculate P(A) and P(B). (4) c. Use also (a) to determine cdf of X. (3) 2. Let f(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere, be the pmf of X. Find a. P(X = 1 or 2), (3) b. P( 1/2 < X < 5/2 ). (3) c. P(1 ? X ? 2). (3) 3. The density function for the random variable X is given by f(x) = { kx for 0 ? x ? 2 0 otherwise a. Find k and P( 1/2 ? x ? 1 ) (4) b. Find F_X(x) (3) 4. A continuous random variable X has pdf f(x) = { kx, 0 ? x ? 2 k(4 – x), 2 ? x ? 4 0, otherwise a. Find the value of k (3) b. Find P( 1/2 ? x ? 2 1/2 ) (3) 5. Let X be a continuous random variable with the pdf f(x) = 2x which has support on the interval (0, 1). Suppose Y = 1 / (1 + X), Compute E(Y). (3) 6. Find the moment generating function of the discrete random variable X which has probability distribution f(x) = 2(1/3)^x, x = 1, 2, 3, ... and use it to determine ?'1 and ?'2. (5) 7. Find the moment generating function of the continuous random variable X whose probability density is given by f(x) = { 1 for 0 < x < 1 0 elsewhere and use it to determine the values of ?'1, ?'2 and ?^2. (5) 8. Show that if a random variable has the probability density function f(x) = 1/2 e^-|x| for -? < x < ? its moment generating function is given by

          26M1/STA21M1
Assignment Marks 50 Due date: 31 March 2023

1. Let X be the sum of the up faces on a roll of a pair of fair 6-sided dice.
a. Determine the pmf of X. (5)
b. Suppose A = {x : x = 7, 11} and B = {x : x = 2, 3, 12}. Then, using the values of f(x) in (a), calculate P(A) and P(B). (4)
c. Use also (a) to determine cdf of X. (3)

2. Let f(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere, be the pmf of X. Find
a. P(X = 1 or 2), (3)
b. P( 1/2 < X < 5/2 ). (3)
c. P(1 ? X ? 2). (3)

3. The density function for the random variable X is given by
f(x) = {
kx for 0 ? x ? 2
0 otherwise

a. Find k and P( 1/2 ? x ? 1 ) (4)
b. Find F_X(x) (3)

4. A continuous random variable X has pdf f(x) = {
kx, 0 ? x ? 2
k(4 – x), 2 ? x ? 4
0, otherwise

a. Find the value of k (3)
b. Find P( 1/2 ? x ? 2 1/2 ) (3)

5. Let X be a continuous random variable with the pdf f(x) = 2x which has support on the interval (0, 1). Suppose Y = 1 / (1 + X), Compute E(Y). (3)

6. Find the moment generating function of the discrete random variable X which has probability distribution f(x) = 2(1/3)^x, x = 1, 2, 3, ...
and use it to determine ?'1 and ?'2. (5)

7. Find the moment generating function of the continuous random variable X whose probability density is given by f(x) = {
1 for 0 < x < 1
0 elsewhere
and use it to determine the values of ?'1, ?'2 and ?^2. (5)

8. Show that if a random variable has the probability density function f(x) = 1/2 e^-|x| for -? < x < ? its moment generating function is given by

26M1/STA21M1
Assignment Marks 50 Due date: 31 March 2023

1. Let X be the sum of the up faces on a roll of a pair of fair 6-sided dice.
a. Determine the pmf of X. (5)
b. Suppose A = x : x = 7, 11 and B = x : x = 2, 3, 12. Then, using the values of f(x) in (a), calculate P(A) and P(B). (4)
c. Use also (a) to determine cdf of X. (3)

2. Let f(x) = x/15, x = 1, 2, 3, 4, 5, zero elsewhere, be the pmf of X. Find
a. P(X = 1 or 2), (3)
b. P( 1/2 < X < 5/2 ). (3)
c. P(1 ? X ? 2). (3)

3. The density function for the random variable X is given by
f(x) =
kx for 0 ? x ? 2
0 otherwise

a. Find k and P( 1/2 ? x ? 1 ) (4)
b. Find FX(x) (3)

4. A continuous random variable X has pdf f(x) =
kx, 0 ? x ? 2
k(4 – x), 2 ? x ? 4
0, otherwise

a. Find the value of k (3)
b. Find P( 1/2 ? x ? 2 1/2 ) (3)

5. Let X be a continuous random variable with the pdf f(x) = 2x which has support on the interval (0, 1). Suppose Y = 1 / (1 + X), Compute E(Y). (3)

6. Find the moment generating function of the discrete random variable X which has probability distribution f(x) = 2(1/3)^x, x = 1, 2, 3, ...
and use it to determine ?'1 and ?'2. (5)

7. Find the moment generating function of the continuous random variable X whose probability density is given by f(x) =
1 for 0 < x < 1
0 elsewhere
and use it to determine the values of ?'1, ?'2 and ?^2. (5)

8. Show that if a random variable has the probability density function f(x) = 1/2 e^-|x| for -? < x < ? its moment generating function is given by

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