IGNOU BMTC 101 2026 - Let E1, E2, E3 and E4 be arbitrary events. Write the followi

Question

Let E₁, E₂, E₃ and E₄ be arbitrary events. Write the following events in set notations:
$\quad$ i) not more than one of E₁, E₂, E₃, E₄.
$\quad$ ii) one and only one of E₁, E₂, E₃, E₄.
$\quad$ iii) E₁ and at least one of E₂, E₃, E₄.
$\quad$ iv) none of E₂, E₃ and E₄ using E₁.
$\quad$ b) Let the probability density function of r.v. X be
$$f(x) = \begin{cases} 1 + x & ; \ -1 < x \le 0 \\ 1 - x & ; \ 0 < x < 1 \\ 0 & ; \ \text{otherwise} \end{cases}$
$\quad$ and if $u = X$ and $v = X^2$ , find Cov(u, v). Also check the independence of u and v.

09 Jan 2026

Answer :

Word Count : 476

i) Not more than one of (E_1, E_2, E_3, E_4) means either zero or one of the events occurs. In set notation: [ (E_1^c \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2 \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3 \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3^c \cap E_4) ] ii) One and only one of (E_1, E_2, E_3, E_4) is exactly the same as the part of (i) excluding the zero-event case: [ (E_1 \cap E_2^c \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2 \cap E_3^c \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3 \cap E_4^c) ;\cup; (E_1^c \cap E_2^c \cap E_3^c \cap E_4) ] iii) (E_1) and at least one of (E_2, E_3, E_4) occurs: [ E_1 \cap (E_2 \cup E_3 \cup E_4) ] iv) None of (E_2, E_3, E_4) occurs, using (E_1): [ E_1 \cap E_2^c \cap E_3^c \cap E_4^c ] --- For part (b), the probability density function is [ f(x) = \begin{cases} 1 + x & -1 < x \le 0 \ 1 - x & 0 < x < 1 \

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