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I guess the Bernoulli random variables are independent.
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edit approved Jul 19, 2017 at 12:03
Synia
edit approved Jul 19, 2017 at 12:03
Synia
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Let $X= \sum_{i=1}^{N} X_i$, where $X_i \sim Bernoulli(p_i)$. Let $Y= \sum_{i=1}^N Y_i$, where $Y_i \sim Bernoulli(p_i+ \delta)$ for some $0 \leq \delta \leq 1- \max_i p_i$. All considered random variables are independent.

Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

Let $X= \sum_{i=1}^{N} X_i$, where $X_i \sim Bernoulli(p_i)$. Let $Y= \sum_{i=1}^N Y_i$, where $Y_i \sim Bernoulli(p_i+ \delta)$ for some $0 \leq \delta \leq 1- \max_i p_i$.

Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

Let $X= \sum_{i=1}^{N} X_i$, where $X_i \sim Bernoulli(p_i)$. Let $Y= \sum_{i=1}^N Y_i$, where $Y_i \sim Bernoulli(p_i+ \delta)$ for some $0 \leq \delta \leq 1- \max_i p_i$. All considered random variables are independent.

Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

added 12 characters in body
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Rakshith
Rakshith
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Let X= \sum_{i=1}^{N} X_i$X= \sum_{i=1}^{N} X_i$, where X_i \sim Bernoulli(p_i)$X_i \sim Bernoulli(p_i)$. Let Y= \sum_{i=1}^N Y_i$Y= \sum_{i=1}^N Y_i$, where Y_i \sim Bernoulli(p_i+ \delta)$Y_i \sim Bernoulli(p_i+ \delta)$ for some 0 \leq \delta \leq 1- \max_i p_i$0 \leq \delta \leq 1- \max_i p_i$.

Can we prove P(X \leq k) \geq P(Y \leq k), \forall k \in {0, \dots, N}$P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

Let X= \sum_{i=1}^{N} X_i, where X_i \sim Bernoulli(p_i). Let Y= \sum_{i=1}^N Y_i, where Y_i \sim Bernoulli(p_i+ \delta) for some 0 \leq \delta \leq 1- \max_i p_i.

Can we prove P(X \leq k) \geq P(Y \leq k), \forall k \in {0, \dots, N}?

Let $X= \sum_{i=1}^{N} X_i$, where $X_i \sim Bernoulli(p_i)$. Let $Y= \sum_{i=1}^N Y_i$, where $Y_i \sim Bernoulli(p_i+ \delta)$ for some $0 \leq \delta \leq 1- \max_i p_i$.

Can we prove $P(X \leq k) \geq P(Y \leq k), \forall k \in \{0, \dots, N\}$?

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Rakshith
Rakshith
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Comparing Cdf of Poisson Binomial distribution for different parameters

Let X= \sum_{i=1}^{N} X_i, where X_i \sim Bernoulli(p_i). Let Y= \sum_{i=1}^N Y_i, where Y_i \sim Bernoulli(p_i+ \delta) for some 0 \leq \delta \leq 1- \max_i p_i.

Can we prove P(X \leq k) \geq P(Y \leq k), \forall k \in {0, \dots, N}?