Random Variables
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Q1
Define mean of a random variable.
Marks:1Answer:
Suppose a random variable X assumes the values x_{1}, x_{2}, ..., x_{n} with probabilities of occurrence p_{1}, p_{2}, ..., p_{n} respectively. Then the mean of the random variable X is defined as
E(x) = x_{1}p_{1} + x_{2}p_{2}+... x_{n}p_{n} = ∑xp.

Q2
Two bad eggs are mixed accidentally with 10 good one. Find the probability distribution of one bad egg in three, drawn at random, from this lot.
Marks:1Answer:

Q3
Check whether the following can be probability distribution of a random variable X:
X: 0 1 2 P(X): 0.3 0.5 0.1 Marks:1Answer:
P(X = 0) + P(X = 1) + P(X = 2) = 0.3 + 0.5 + 0.1
= 0.9 ≠ 1Hence, the given distribution of probabilities is not a probability distribution.

Q4
Define random variable.
Marks:1Answer:
A random variable is a real valued function having domain as the sample space associated with a random experiment.

Q5
If X is a random variable and k is a constant then prove that
Var(kX) = k^{2}VarX.
Marks:1Answer:
Var(X) = E(X^{2}) – [E(X)]^{2}
Var(kX) = E(k^{2}X^{2}) – [E(kX)]^{2}
= k^{2}E(X^{2}) – [kE(X)]^{2}
= k^{2}E(X^{2}) – k^{2}[E(X)]^{2}
= k^{2}[E(X^{2}) – {E(X)^{2}}]
= k^{2}Var(X)